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<span class="kw">let </span><font color="Maroon" title="c1">L</font> be  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a>  <a href="orders_2.html#NM1" title="ORDERS_2:NM.1">Poset</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">R</font> being  <a href="waybel35.html#V1" title="WAYBEL35:attr.1">extra-order</a>  <a href="alg_1.html#NM1" title="ALG_1:NM.1">Relation</a> of <font color="Maroon" title="c1">L</font><br/> for <font color="Olive" title="b2">C</font> being  <a href="waybel35.html#V3" title="WAYBEL35:attr.3">satisfying_SIC</a>   <a href="waybel35.html#M1" title="WAYBEL35:mode.1">strict_chain</a> of <font color="Olive" title="b1">R</font><br/> for <font color="Olive" title="b3">p</font>, <font color="Olive" title="b4">q</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">C</font> &amp; <font color="Olive" title="b4">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">C</font> &amp; <font color="Olive" title="b3">p</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Olive" title="b4">q</font> holds <br/> ex <font color="Olive" title="b5">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b3">p</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Olive" title="b5">y</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b5">y</font>,<font color="Olive" title="b4">q</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">R</font> &amp; <font color="Olive" title="b5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><a href="waybel35.html#K4" title="WAYBEL35:func.4">SetBelow</a> <font color="Olive" title="b1">R</font>,<font color="Olive" title="b2">C</font>,<font color="Olive" title="b5">y</font></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">R</font> be   <a href="waybel35.html#V1" title="WAYBEL35:attr.1">extra-order</a>  <a href="alg_1.html#NM1" title="ALG_1:NM.1">Relation</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"> for <font color="Olive" title="b1">C</font> being  <a href="waybel35.html#V3" title="WAYBEL35:attr.3">satisfying_SIC</a>   <a href="waybel35.html#M1" title="WAYBEL35:mode.1">strict_chain</a> of <font color="Maroon" title="c2">R</font><br/> for <font color="Olive" title="b2">p</font>, <font color="Olive" title="b3">q</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> &amp; <font color="Olive" title="b3">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> &amp; <font color="Olive" title="b2">p</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Olive" title="b3">q</font> holds <br/> ex <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">L</font> st <br/>( <font color="Olive" title="b2">p</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Olive" title="b4">y</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b4">y</font>,<font color="Olive" title="b3">q</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> &amp; <font color="Olive" title="b4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><a href="waybel35.html#K4" title="WAYBEL35:func.4">SetBelow</a> <font color="Maroon" title="c2">R</font>,<font color="Olive" title="b1">C</font>,<font color="Olive" title="b4">y</font></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">C</font> be   <a href="waybel35.html#V3" title="WAYBEL35:attr.3">satisfying_SIC</a>   <a href="waybel35.html#M1" title="WAYBEL35:mode.1">strict_chain</a> of <font color="Maroon" title="c2">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">p</font>, <font color="Olive" title="b2">q</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font> &amp; <font color="Olive" title="b2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font> &amp; <font color="Olive" title="b1">p</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Olive" title="b2">q</font> holds <br/> ex <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">L</font> st <br/>( <font color="Olive" title="b1">p</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Olive" title="b3">y</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b3">y</font>,<font color="Olive" title="b2">q</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> &amp; <font color="Olive" title="b3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><a href="waybel35.html#K4" title="WAYBEL35:func.4">SetBelow</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">C</font>,<font color="Olive" title="b3">y</font></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">p</font>, <font color="Maroon" title="c5">q</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font> &amp; <font color="Maroon" title="c5">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font> &amp; <font color="Maroon" title="c4">p</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c5">q</font> implies  ex <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">L</font> st <br/>( <font color="Maroon" title="c4">p</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Olive" title="b1">y</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">y</font>,<font color="Maroon" title="c5">q</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> &amp; <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><a href="waybel35.html#K4" title="WAYBEL35:func.4">SetBelow</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">C</font>,<font color="Olive" title="b1">y</font></span>)</span> ) )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:50"/><span class="lab"><font color="Green" title="E22">A1</font></span>: 
<font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font>
 <span class="kw">and </span><br/><a NAME="E2:50"/><span class="lab"><font color="Green" title="E23">A2</font></span>: 
<font color="Maroon" title="c5">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font>
 <span class="kw">and </span><br/><a NAME="E3:50"/><span class="lab"><font color="Green" title="E24">A3</font></span>: 
<font color="Maroon" title="c4">p</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c5">q</font>
 ; <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" title="b1">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Maroon" title="c4">p</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Olive" title="b1">y</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">y</font>,<font color="Maroon" title="c5">q</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> &amp; <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><a href="waybel35.html#K4" title="WAYBEL35:func.4">SetBelow</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">C</font>,<font color="Olive" title="b1">y</font></span>)</span> )</span><br/>

<a NAME="E4:50"/><span class="lab"><font color="Green" title="E25">A4</font></span>: 
<font color="Maroon" title="c2">R</font> <a href="waybel35.html#R1" title="WAYBEL35:pred.1">satisfies_SIC_on</a> <font color="Maroon" title="c3">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel35.html#D7" target="_self" title="WAYBEL35:def.7">Def7</a></span>;<br/>
<a NAME="E5:50"/>
not <font color="Maroon" title="c5">q</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:50"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="ref" href="orders_2.html#T30" title="ORDERS_2:th.30">ORDERS_2:30</a></span>;<br/>
<a NAME="E6:50"/><span class="kw">then </span>
not <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c4">p</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_4.html#D4" title="WAYBEL_4:def.4">WAYBEL_4:def 4</a></span>;<br/>
<a NAME="E7:50"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E2:50"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="txt" href="#E3:50"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="ref" href="waybel35.html#D3" target="_self" title="WAYBEL35:def.3">Def3</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">w</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><span class="kw"> such that </span><br/><a NAME="E9:50"/><span class="lab"><font color="Green" title="E26">A5</font></span>: 
<font color="Maroon" title="c6">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font>
 <span class="kw">and </span><br/><a NAME="E10:50"/><span class="lab"><font color="Green" title="E27">A6</font></span>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c6">w</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font>
 <span class="kw">and </span><br/><a NAME="E11:50"/><span class="lab"><font color="Green" title="E28">A7</font></span>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">w</font>,<font color="Maroon" title="c5">q</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font>
 <span class="kw">and </span><br/><a NAME="E12:50"/><span class="lab"><font color="Green" title="E29">A8</font></span>: 
<font color="Maroon" title="c4">p</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c6">w</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E2:50"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="txt" href="#E3:50"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="txt" href="#E4:50"><span class="lab"><font color="Green" title="E25">A4</font></span></a>, <a class="ref" href="waybel35.html#T16" target="_self" title="WAYBEL35:th.16">Th16</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">x1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><span class="kw"> such that </span><br/><a NAME="E14:50"/><span class="lab"><font color="Green" title="E30">A9</font></span>: 
<font color="Maroon" title="c7">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font>
 <span class="kw">and </span><br/><a NAME="E15:50"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c7">x1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font>
 <span class="kw">and </span><br/><a NAME="E16:50"/><span class="lab"><font color="Green" title="E31">A10</font></span>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">x1</font>,<font color="Maroon" title="c6">w</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font>
 <span class="kw">and </span><br/><a NAME="E17:50"/><span class="lab"><font color="Green" title="E32">A11</font></span>: 
<font color="Maroon" title="c4">p</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c7">x1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E4:50"><span class="lab"><font color="Green" title="E25">A4</font></span></a>, <a class="txt" href="#E9:50"><span class="lab"><font color="Green" title="E26">A5</font></span></a>, <a class="txt" href="#E10:50"><span class="lab"><font color="Green" title="E27">A6</font></span></a>, <a class="txt" href="#E12:50"><span class="lab"><font color="Green" title="E29">A8</font></span></a>, <a class="ref" href="waybel35.html#T16" target="_self" title="WAYBEL35:th.16">Th16</a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp; $3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> &amp; $3 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">$2,$3</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> &amp; <font color="Olive" title="b2">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">w</font> );<br/>
<a NAME="E18:50"/><span class="lab"><font color="Green" title="E33">A12</font></span>: 
<font color="Maroon" title="c5">q</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">q</font>
 
;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">D</font> =  <a href="waybel35.html#K4" title="WAYBEL35:func.4">SetBelow</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c6">w</font> as  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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:50"><span class="lab"><font color="Green" title="E30">A9</font></span></a>, <a class="txt" href="#E16:50"><span class="lab"><font color="Green" title="E31">A10</font></span></a>, <a class="ref" href="waybel35.html#T18" target="_self" title="WAYBEL35:th.18">Th18</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">g</font> = <font color="Maroon" title="c7">x1</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c8">D</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:50"><span class="lab"><font color="Green" title="E30">A9</font></span></a>, <a class="txt" href="#E16:50"><span class="lab"><font color="Green" title="E31">A10</font></span></a>, <a class="ref" href="waybel35.html#T18" target="_self" title="WAYBEL35:th.18">Th18</a></span>;<br/>
<a NAME="E21:50"/><span class="lab"><font color="Green" title="E34">A13</font></span>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c8">D</font>  ex <font color="Olive" title="b3">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c8">D</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</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">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c8">D</font>  ex <font color="Olive" title="b2">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c8">D</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">n</font>,<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>]</span><br/><span class="kw">let </span><font color="Maroon" title="c11">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c8">D</font>; <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" title="b1">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c8">D</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">n</font>,<font color="Maroon" title="c11">x</font>,<font color="Olive" title="b1">y</font>]</span><br/>



<a NAME="E1:50_1"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">D</font>
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">t</font> = <font color="Maroon" title="c11">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<a NAME="E3:50_1"/><span class="lab"><font color="Green" title="E35">A14</font></span>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel35.html#T18" target="_self" title="WAYBEL35:th.18">Th18</a></span>;<br/>
<a NAME="E4:50_1"/><span class="lab"><font color="Green" title="E36">A15</font></span>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c6">w</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel35.html#T18" target="_self" title="WAYBEL35:th.18">Th18</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:50_1_1"/>
( <font color="Maroon" title="c11">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">w</font> or <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">w</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:50_1_1_1"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">w</font>
 ; <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" title="b1">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c8">D</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">n</font>,<font color="Maroon" title="c11">x</font>,<font color="Olive" title="b1">y</font>]</span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><span class="kw"> such that </span><br/><a NAME="E3:50_1_1_1"/><span class="lab"><font color="Green" title="E37">A16</font></span>: 
<font color="Maroon" title="c13">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font>
 <span class="kw">and </span><br/><a NAME="E4:50_1_1_1"/><span class="lab"><font color="Green" title="E38">A17</font></span>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c13">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font>
 <span class="kw">and </span><br/><a NAME="E5:50_1_1_1"/><span class="lab"><font color="Green" title="E39">A18</font></span>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c13">b</font>,<font color="Maroon" title="c6">w</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font>
 <span class="kw">and </span><br/><a NAME="E6:50_1_1_1"/>
<font color="Maroon" title="c12">t</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c13">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:50"><span class="lab"><font color="Green" title="E25">A4</font></span></a>, <a class="txt" href="#E9:50"><span class="lab"><font color="Green" title="E26">A5</font></span></a>, <a class="txt" href="#E3:50_1"><span class="lab"><font color="Green" title="E35">A14</font></span></a>, <a class="txt" href="#E4:50_1"><span class="lab"><font color="Green" title="E36">A15</font></span></a>, <a class="ref" href="waybel35.html#T16" target="_self" title="WAYBEL35:th.16">Th16</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c14">y</font> = <font color="Maroon" title="c13">b</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c8">D</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:50_1_1_1"><span class="lab"><font color="Green" title="E37">A16</font></span></a>, <a class="txt" href="#E5:50_1_1_1"><span class="lab"><font color="Green" title="E39">A18</font></span></a>, <a class="ref" href="waybel35.html#T18" target="_self" title="WAYBEL35:th.18">Th18</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c14">y</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">S<sub>1</sub></font>[<font color="Maroon" title="c10">n</font>,<font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c14">y</font>]</span><br/><span class="kw">take </span>
<font color="Maroon" title="c12">t</font>
; <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" title="b1">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">t</font> &amp; <font color="Maroon" title="c14">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> &amp; <font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c14">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> &amp; <font color="Olive" title="b1">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">w</font> )</span><br/><span class="kw">take </span>
<font color="Maroon" title="c13">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="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">t</font> &amp; <font color="Maroon" title="c14">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">b</font> &amp; <font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c14">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> &amp; <font color="Maroon" title="c13">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">w</font> )</span><br/><span class="kw">thus </span><a NAME="E8:50_1_1_1"/>
( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">t</font> &amp; <font color="Maroon" title="c14">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">b</font> &amp; <font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c14">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> &amp; <font color="Maroon" title="c13">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">w</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:50_1_1_1"><span class="lab"><font color="Green" title="E37">A16</font></span></a>, <a class="txt" href="#E4:50_1_1_1"><span class="lab"><font color="Green" title="E38">A17</font></span></a>, <a class="txt" href="#E5:50_1_1_1"><span class="lab"><font color="Green" title="E39">A18</font></span></a>, <a class="ref" href="waybel_4.html#D4" title="WAYBEL_4:def.4">WAYBEL_4:def 4</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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:50_1_1_2"/><span class="lab"><font color="Green" title="E37">A19</font></span>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">w</font>
 ; <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" title="b1">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c8">D</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">n</font>,<font color="Maroon" title="c11">x</font>,<font color="Olive" title="b1">y</font>]</span><br/><div class="add"><span class="kw">take </span>
<font color="Maroon" title="c11">x</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">S<sub>1</sub></font>[<font color="Maroon" title="c10">n</font>,<font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c11">x</font>]</span><br/><span class="kw">take </span>
<font color="Maroon" title="c12">t</font>
; <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" title="b1">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">t</font> &amp; <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> &amp; <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c11">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> &amp; <font color="Olive" title="b1">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">w</font> )</span><br/><span class="kw">take </span>
<font color="Maroon" title="c12">t</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="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">t</font> &amp; <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">t</font> &amp; <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c11">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> &amp; <font color="Maroon" title="c12">t</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">w</font> )</span><br/><span class="kw">thus </span><a NAME="E2:50_1_1_2"/>
( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">t</font> &amp; <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">t</font> &amp; <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c11">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> &amp; <font color="Maroon" title="c12">t</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">w</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50_1_1_2"><span class="lab"><font color="Green" title="E37">A19</font></span></a>, <a class="ref" href="waybel35.html#T18" target="_self" title="WAYBEL35:th.18">Th18</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/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c10">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<font color="Maroon" title="c8">D</font><span class="kw"> such that </span><br/><a NAME="E23:50"/><span class="lab"><font color="Green" title="E35">A20</font></span>: 
<font color="Maroon" title="c10">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">g</font>
 <span class="kw">and </span><br/><a NAME="E24:50"/><span class="lab"><font color="Green" title="E36">A21</font></span>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Maroon" title="c10">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font>,<font color="Maroon" title="c10">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="recdef_1.html#S2" title="RECDEF_1:sch.2">RECDEF_1:sch 2</a>(<a class="txt" href="#E21:50"><span class="lab"><font color="Green" title="E34">A13</font></span></a>);<br/></span>
<span class="kw">reconsider </span><font color="Maroon" title="c11">f</font> = <font color="Maroon" title="c10">f</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</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">by</span> <span class="lab"><a class="ref" href="funct_2.html#T9" title="FUNCT_2:th.9">FUNCT_2:9</a></span>;<br/>
<span class="kw">take </span><font color="Maroon" title="c12">y</font> =  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c11">f</font></span>)</span>; <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">p</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c12">y</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c5">q</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> &amp; <font color="Maroon" title="c12">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><a href="waybel35.html#K4" title="WAYBEL35:func.4">SetBelow</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c12">y</font></span>)</span> )</span><br/>

<a NAME="E26:50"/><span class="lab"><font color="Green" title="E37">A22</font></span>: 
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c11">f</font>,<font color="Maroon" title="c1">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T17" title="YELLOW_0:th.17">YELLOW_0:17</a></span>;<br/>
<a NAME="E27:50"/><span class="lab"><font color="Green" title="E38">A23</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c11">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E28:50"/><span class="kw">then </span>
<font color="Maroon" title="c7">x1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c12">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:50"><span class="lab"><font color="Green" title="E35">A20</font></span></a>, <a class="txt" href="#E26:50"><span class="lab"><font color="Green" title="E37">A22</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a>, <a class="ref" href="yellow_4.html#T1" title="YELLOW_4:th.1">YELLOW_4:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E29:50"/>
<font color="Maroon" title="c4">p</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c12">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:50"><span class="lab"><font color="Green" title="E32">A11</font></span></a>, <a class="ref" href="orders_2.html#T32" title="ORDERS_2:th.32">ORDERS_2:32</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c5">q</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> &amp; <font color="Maroon" title="c12">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><a href="waybel35.html#K4" title="WAYBEL35:func.4">SetBelow</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c12">y</font></span>)</span> )</span><br/>

<a NAME="E30:50"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c11">f</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Maroon" title="c6">w</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="lattice3.html#D9" title="LATTICE3:def.9">LATTICE3:def 9</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c13">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="c11">f</font> or <font color="Maroon" title="c13">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">w</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:50_2"/>
<font color="Maroon" title="c13">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="c11">f</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="c13">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">w</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">n</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:50_2"/><span class="lab"><font color="Green" title="E39">A24</font></span>: 
<font color="Maroon" title="c14">n</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="c11">f</font>
 <span class="kw">and </span><br/><a NAME="E4:50_2"/><span class="lab"><font color="Green" title="E40">A25</font></span>: 
<font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">x</font>
 <span class="kw">by</span> <span class="lab"><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="c15">n</font> = <font color="Maroon" title="c14">n</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:50_2"><span class="lab"><font color="Green" title="E39">A24</font></span></a></span>;<br/>
<a NAME="E6:50_2"/><span class="lab"><font color="Green" title="E41">A26</font></span>: 
 ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Maroon" title="c11">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp; <font color="Maroon" title="c11">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> &amp; <font color="Maroon" title="c11">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> &amp; <font color="Olive" title="b2">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">w</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:50"><span class="lab"><font color="Green" title="E36">A21</font></span></a></span>;<br/>
<a NAME="E7:50_2"/><span class="kw">then </span>
<font color="Maroon" title="c13">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c11">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:50_2"><span class="lab"><font color="Green" title="E40">A25</font></span></a>, <a class="ref" href="waybel_4.html#D4" title="WAYBEL_4:def.4">WAYBEL_4:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:50_2"/>
<font color="Maroon" title="c13">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">w</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:50_2"><span class="lab"><font color="Green" title="E41">A26</font></span></a>, <a class="ref" href="orders_2.html#T26" title="ORDERS_2:th.26">ORDERS_2:26</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/>


</div><span class="kw">end;</span></div>
<a NAME="E31:50"/><span class="kw">then </span>
<font color="Maroon" title="c12">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">w</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:50"><span class="lab"><font color="Green" title="E37">A22</font></span></a>, <a class="ref" href="yellow_0.html#D9" title="YELLOW_0:def.9">YELLOW_0:def 9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E32:50"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c5">q</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:50"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="txt" href="#E18:50"><span class="lab"><font color="Green" title="E33">A12</font></span></a>, <a class="ref" href="waybel_4.html#D5" title="WAYBEL_4:def.5">WAYBEL_4:def 5</a></span>; <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="c12">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><a href="waybel35.html#K4" title="WAYBEL35:func.4">SetBelow</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c12">y</font></span>)</span></span><br/>

<a NAME="E33:50"/><span class="lab"><font color="Green" title="E39">A27</font></span>: 
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a>  <a href="waybel35.html#K4" title="WAYBEL35:func.4">SetBelow</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c1">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T17" title="YELLOW_0:th.17">YELLOW_0:17</a></span>;<br/>
<a NAME="E34:50"/><span class="lab"><font color="Green" title="E40">A28</font></span>: 
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st  <a href="waybel35.html#K4" title="WAYBEL35:func.4">SetBelow</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c12">y</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive" title="b1">x</font> holds <br/><font color="Maroon" title="c12">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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"> (  <a href="waybel35.html#K4" title="WAYBEL35:func.4">SetBelow</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c12">y</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Maroon" title="c13">x</font> implies <font color="Maroon" title="c12">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c13">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:50_3"/><span class="lab"><font color="Green" title="E41">A29</font></span>: 
 <a href="waybel35.html#K4" title="WAYBEL35:func.4">SetBelow</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c12">y</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Maroon" title="c13">x</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="c12">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c13">x</font></span><br/>

<a NAME="E2:50_3"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c11">f</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Maroon" title="c13">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span><font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font>;<br/>
<span class="kw">let </span><font color="Maroon" title="c14">m</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="lattice3.html#D9" title="LATTICE3:def.9">LATTICE3:def 9</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c14">m</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="c11">f</font> or <font color="Maroon" title="c14">m</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c13">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:50_3_1"/>
<font color="Maroon" title="c14">m</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="c11">f</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="c14">m</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c13">x</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">n</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:50_3_1"/><span class="lab"><font color="Green" title="E42">A30</font></span>: 
<font color="Maroon" title="c15">n</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="c11">f</font>
 <span class="kw">and </span><br/><a NAME="E4:50_3_1"/><span class="lab"><font color="Green" title="E43">A31</font></span>: 
<font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">m</font>
 <span class="kw">by</span> <span class="lab"><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="c16">n</font> = <font color="Maroon" title="c15">n</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:50_3_1"><span class="lab"><font color="Green" title="E42">A30</font></span></a></span>;<br/>
<a NAME="E6:50_3_1"/><span class="lab"><font color="Green" title="E44">A32</font></span>: 
<font color="Maroon" title="c11">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">n</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c11">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">n</font>
 
;<br/>
<a NAME="E7:50_3_1"/><span class="lab"><font color="Green" title="E45">A33</font></span>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_3_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c17">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</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">S<sub>2</sub></font>[<font color="Maroon" title="c17">k</font>] implies <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c17">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<a NAME="E1:50_3_1_1"/>
<font color="Maroon" title="c17">k</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/>
<a NAME="E2:50_3_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp; <font color="Maroon" title="c11">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> &amp; <font color="Maroon" title="c11">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> &amp; <font color="Olive" title="b2">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">w</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:50"><span class="lab"><font color="Green" title="E36">A21</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:50_3_1_1"/>
( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c17">k</font>] implies <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c17">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )
 ; <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>
<a NAME="E8:50_3_1"/><span class="lab"><font color="Green" title="E46">A34</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:50"><span class="lab"><font color="Green" title="E30">A9</font></span></a>, <a class="txt" href="#E23:50"><span class="lab"><font color="Green" title="E35">A20</font></span></a></span>;<br/>
<a NAME="E9:50_3_1"/>
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="#E8:50_3_1"><span class="lab"><font color="Green" title="E46">A34</font></span></a>, <a class="txt" href="#E7:50_3_1"><span class="lab"><font color="Green" title="E45">A33</font></span></a>);<br/></span>
<a NAME="E10:50_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A35</font></span>: 
<font color="Maroon" title="c11">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font>
 
;<br/>
<a NAME="E11:50_3_1"/><span class="lab"><font color="Green" title="E48">A36</font></span>: 
 ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Maroon" title="c11">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp; <font color="Maroon" title="c11">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> &amp; <font color="Maroon" title="c11">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> &amp; <font color="Olive" title="b2">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">w</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:50"><span class="lab"><font color="Green" title="E36">A21</font></span></a></span>;<br/>
<a NAME="E12:50_3_1"/>
<font color="Maroon" title="c11">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c12">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:50"><span class="lab"><font color="Green" title="E37">A22</font></span></a>, <a class="txt" href="#E27:50"><span class="lab"><font color="Green" title="E38">A23</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a>, <a class="ref" href="yellow_4.html#T1" title="YELLOW_4:th.1">YELLOW_4:1</a></span>;<br/>
<a NAME="E13:50_3_1"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c14">m</font>,<font color="Maroon" title="c12">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:50_3_1"><span class="lab"><font color="Green" title="E43">A31</font></span></a>, <a class="txt" href="#E11:50_3_1"><span class="lab"><font color="Green" title="E48">A36</font></span></a>, <a class="txt" href="#E6:50_3_1"><span class="lab"><font color="Green" title="E44">A32</font></span></a>, <a class="ref" href="waybel_4.html#D5" title="WAYBEL_4:def.5">WAYBEL_4:def 5</a></span>;<br/>
<a NAME="E14:50_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel35.html#K4" title="WAYBEL35:func.4">SetBelow</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c12">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:50_3_1"><span class="lab"><font color="Green" title="E43">A31</font></span></a>, <a class="txt" href="#E10:50_3_1"><span class="lab"><font color="Green" title="E47">A35</font></span></a>, <a class="ref" href="waybel35.html#T18" target="_self" title="WAYBEL35:th.18">Th18</a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:50_3_1"/>
<font color="Maroon" title="c14">m</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c13">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50_3"><span class="lab"><font color="Green" title="E41">A29</font></span></a>, <a class="ref" href="lattice3.html#D9" title="LATTICE3:def.9">LATTICE3:def 9</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/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:50_3"/>
<font color="Maroon" title="c12">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c13">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:50"><span class="lab"><font color="Green" title="E37">A22</font></span></a>, <a class="ref" href="yellow_0.html#D9" title="YELLOW_0:def.9">YELLOW_0:def 9</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/>


</div><span class="kw">end;</span></div>
<a NAME="E35:50"/>
 <a href="waybel35.html#K4" title="WAYBEL35:func.4">SetBelow</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c12">y</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Maroon" title="c12">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel35.html#T19" target="_self" title="WAYBEL35:th.19">Th19</a></span>;<br/>
<span class="kw">hence </span><a NAME="E36:50"/>
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><a href="waybel35.html#K4" title="WAYBEL35:func.4">SetBelow</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c12">y</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:50"><span class="lab"><font color="Green" title="E40">A28</font></span></a>, <a class="txt" href="#E33:50"><span class="lab"><font color="Green" title="E39">A27</font></span></a>, <a class="ref" href="yellow_0.html#D9" title="YELLOW_0:def.9">YELLOW_0:def 9</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/>


</div>
