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

<span class="kw">let </span><font color="Maroon" title="c1">L</font> be   <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <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 <font color="Maroon" title="c1">L</font> is  <a href="yellow11.html#V1" title="YELLOW11:attr.1">modular</a>  holds <br/> <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p1"><a href="yellow11.html#K3" title="YELLOW11:func.3">[#</a><span class="default"><font color="Olive" title="b2">b</font>,<span class="p2">(<span class="default"><font color="Olive" title="b1">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Olive" title="b2">b</font></span>)</span></span><a href="yellow11.html#K3" title="YELLOW11:func.3">#]</a></span>, <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p1"><a href="yellow11.html#K3" title="YELLOW11:func.3">[#</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Olive" title="b2">b</font></span>)</span>,<font color="Olive" title="b1">a</font></span><a href="yellow11.html#K3" title="YELLOW11:func.3">#]</a></span> <a href="waybel_1.html#R5" title="WAYBEL_1:pred.5">are_isomorphic</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">L</font> is  <a href="yellow11.html#V1" title="YELLOW11:attr.1">modular</a>  implies  <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p1"><a href="yellow11.html#K3" title="YELLOW11:func.3">[#</a><span class="default"><font color="Maroon" title="c3">b</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">b</font></span>)</span></span><a href="yellow11.html#K3" title="YELLOW11:func.3">#]</a></span>, <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p1"><a href="yellow11.html#K3" title="YELLOW11:func.3">[#</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c3">b</font></span>)</span>,<font color="Maroon" title="c2">a</font></span><a href="yellow11.html#K3" title="YELLOW11:func.3">#]</a></span> <a href="waybel_1.html#R5" title="WAYBEL_1:pred.5">are_isomorphic</a>  )</span><br/>



<span class="kw">assume </span><a NAME="E1:28"/><span class="lab"><font color="Green" title="E14">A1</font></span>: 
<font color="Maroon" title="c1">L</font> is  <a href="yellow11.html#V1" title="YELLOW11:attr.1">modular</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p1"><a href="yellow11.html#K3" title="YELLOW11:func.3">[#</a><span class="default"><font color="Maroon" title="c3">b</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">b</font></span>)</span></span><a href="yellow11.html#K3" title="YELLOW11:func.3">#]</a></span>, <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p1"><a href="yellow11.html#K3" title="YELLOW11:func.3">[#</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c3">b</font></span>)</span>,<font color="Maroon" title="c2">a</font></span><a href="yellow11.html#K3" title="YELLOW11:func.3">#]</a></span> <a href="waybel_1.html#R5" title="WAYBEL_1:pred.5">are_isomorphic</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">object</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ] <span class="kw">means </span>( $2 is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> &amp; (  for <font color="Olive" title="b1">X</font>, <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">L</font>  st $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">X</font> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">Y</font> holds <br/><font color="Olive" title="b2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">X</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c2">a</font> ) );<br/>
<a NAME="E2:28"/><span class="lab"><font color="Green" title="E15">A2</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2"><a href="yellow11.html#K3" title="YELLOW11:func.3">[#</a><span class="default"><font color="Maroon" title="c3">b</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">b</font></span>)</span></span><a href="yellow11.html#K3" title="YELLOW11:func.3">#]</a></span></span>)</span> holds <br/> ex <font color="Olive" title="b2">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2"><a href="yellow11.html#K3" title="YELLOW11:func.3">[#</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c3">b</font></span>)</span>,<font color="Maroon" title="c2">a</font></span><a href="yellow11.html#K3" title="YELLOW11:func.3">#]</a></span></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="28_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2"><a href="yellow11.html#K3" title="YELLOW11:func.3">[#</a><span class="default"><font color="Maroon" title="c3">b</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">b</font></span>)</span></span><a href="yellow11.html#K3" title="YELLOW11:func.3">#]</a></span></span>)</span> implies  ex <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2"><a href="yellow11.html#K3" title="YELLOW11:func.3">[#</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c3">b</font></span>)</span>,<font color="Maroon" title="c2">a</font></span><a href="yellow11.html#K3" title="YELLOW11:func.3">#]</a></span></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">x</font>,<font color="Olive" title="b1">y</font>] ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:28_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2"><a href="yellow11.html#K3" title="YELLOW11:func.3">[#</a><span class="default"><font color="Maroon" title="c3">b</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">b</font></span>)</span></span><a href="yellow11.html#K3" title="YELLOW11:func.3">#]</a></span></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2"><a href="yellow11.html#K3" title="YELLOW11:func.3">[#</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c3">b</font></span>)</span>,<font color="Maroon" title="c2">a</font></span><a href="yellow11.html#K3" title="YELLOW11:func.3">#]</a></span></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">x</font>,<font color="Olive" title="b1">y</font>] )</span><br/>

<a NAME="E2:28_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A3</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="yellow11.html#K3" title="YELLOW11:func.3">[#</a><span class="default"><font color="Maroon" title="c3">b</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">b</font></span>)</span></span><a href="yellow11.html#K3" title="YELLOW11:func.3">#]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#D15" onmouseover="rs('yellow_0/D15')" onmouseout="rh()">YELLOW_0:def 15</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">x1</font> = <font color="Maroon" title="c4">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/>
<span class="kw">take </span><font color="Maroon" title="c6">y</font> = <font color="Maroon" title="c2">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c5">x1</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2"><a href="yellow11.html#K3" title="YELLOW11:func.3">[#</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c3">b</font></span>)</span>,<font color="Maroon" title="c2">a</font></span><a href="yellow11.html#K3" title="YELLOW11:func.3">#]</a></span></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c6">y</font>] )</span><br/>

<a NAME="E4:28_1"/>
<font color="Maroon" title="c5">x1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:28_1"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="ref" href="yellow11.html#D4" target="_self" onmouseover="rs('yellow11/D4')" onmouseout="rh()">Def4</a></span>;<br/>
<a NAME="E5:28_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">b</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_5.html#T6" onmouseover="rs('yellow_5/T6')" onmouseout="rh()">YELLOW_5:6</a></span>;<br/>
<a NAME="E6:28_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A4</font></span>: 
<font color="Maroon" title="c6">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#T18" onmouseover="rs('lattice3/T18')" onmouseout="rh()">LATTICE3:18</a></span>;<br/>
<a NAME="E7:28_1"/>
<font color="Maroon" title="c3">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">x1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:28_1"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="ref" href="yellow11.html#D4" target="_self" onmouseover="rs('yellow11/D4')" onmouseout="rh()">Def4</a></span>;<br/>
<a NAME="E8:28_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c3">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_5.html#T6" onmouseover="rs('yellow_5/T6')" onmouseout="rh()">YELLOW_5:6</a></span>;<br/>
<a NAME="E9:28_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="yellow11.html#K3" title="YELLOW11:func.3">[#</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c3">b</font></span>)</span>,<font color="Maroon" title="c2">a</font></span><a href="yellow11.html#K3" title="YELLOW11:func.3">#]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:28_1"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="ref" href="yellow11.html#D4" target="_self" onmouseover="rs('yellow11/D4')" onmouseout="rh()">Def4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:28_1"/>
<font color="Maroon" title="c6">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2"><a href="yellow11.html#K3" title="YELLOW11:func.3">[#</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c3">b</font></span>)</span>,<font color="Maroon" title="c2">a</font></span><a href="yellow11.html#K3" title="YELLOW11:func.3">#]</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#D15" onmouseover="rs('yellow_0/D15')" onmouseout="rh()">YELLOW_0:def 15</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c6">y</font>]</span><br/>

<span class="kw">thus </span><a NAME="E11:28_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c6">y</font>]
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c4">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2"><a href="yellow11.html#K3" title="YELLOW11:func.3">[#</a><span class="default"><font color="Maroon" title="c3">b</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">b</font></span>)</span></span><a href="yellow11.html#K3" title="YELLOW11:func.3">#]</a></span></span>)</span>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2"><a href="yellow11.html#K3" title="YELLOW11:func.3">[#</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c3">b</font></span>)</span>,<font color="Maroon" title="c2">a</font></span><a href="yellow11.html#K3" title="YELLOW11:func.3">#]</a></span></span>)</span><span class="kw"> such that </span><br/><a NAME="E4:28"/><span class="lab"><font color="Green" title="E16">A5</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2"><a href="yellow11.html#K3" title="YELLOW11:func.3">[#</a><span class="default"><font color="Maroon" title="c3">b</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">b</font></span>)</span></span><a href="yellow11.html#K3" title="YELLOW11:func.3">#]</a></span></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S1" onmouseover="rs('funct_2/S1')" onmouseout="rh()">FUNCT_2:sch 1</a>(<a class="txt" href="#E2:28"><span class="lab"><font color="Green" title="E15">A2</font></span></a>);<br/></span>
<span class="kw">reconsider </span><font color="Maroon" title="c5">f</font> = <font color="Maroon" title="c4">f</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2"><a href="yellow11.html#K3" title="YELLOW11:func.3">[#</a><span class="default"><font color="Maroon" title="c3">b</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">b</font></span>)</span></span><a href="yellow11.html#K3" title="YELLOW11:func.3">#]</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2"><a href="yellow11.html#K3" title="YELLOW11:func.3">[#</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c3">b</font></span>)</span>,<font color="Maroon" title="c2">a</font></span><a href="yellow11.html#K3" title="YELLOW11:func.3">#]</a></span></span>)</span> ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c5">f</font>
; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="waybel_1.html#D8" onmouseover="rs('waybel_1/D8')" onmouseout="rh()">WAYBEL_1:def 8</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">f</font> is  <a href="waybel_0.html#V23" title="WAYBEL_0:attr.23">isomorphic</a> </span><br/>

<span class="kw">thus </span><a NAME="E6:28"/>
<font color="Maroon" title="c5">f</font> is  <a href="waybel_0.html#V23" title="WAYBEL_0:attr.23">isomorphic</a> 
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="28_2"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:28_2"/><span class="lab"><font color="Green" title="E17">A6</font></span>: 
<font color="Maroon" title="c3">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T22" onmouseover="rs('yellow_0/T22')" onmouseout="rh()">YELLOW_0:22</a></span>;<br/>
<a NAME="E2:28_2"/>
<font color="Maroon" title="c3">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">b</font>
 
;<br/>
<a NAME="E3:28_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="yellow11.html#K3" title="YELLOW11:func.3">[#</a><span class="default"><font color="Maroon" title="c3">b</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">b</font></span>)</span></span><a href="yellow11.html#K3" title="YELLOW11:func.3">#]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28_2"><span class="lab"><font color="Green" title="E17">A6</font></span></a>, <a class="ref" href="yellow11.html#D4" target="_self" onmouseover="rs('yellow11/D4')" onmouseout="rh()">Def4</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">s1</font> =  <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p1"><a href="yellow11.html#K3" title="YELLOW11:func.3">[#</a><span class="default"><font color="Maroon" title="c3">b</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c2">a</font></span>)</span></span><a href="yellow11.html#K3" title="YELLOW11:func.3">#]</a></span> as   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="yellow_0.html#V4" title="YELLOW_0:attr.4">full</a>  <a href="lattice5.html#NM2" title="LATTICE5:NM.2">Sublattice</a> of <font color="Maroon" title="c1">L</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#D15" onmouseover="rs('yellow_0/D15')" onmouseout="rh()">YELLOW_0:def 15</a></span>;<br/>
<a NAME="E5:28_2"/><span class="lab"><font color="Green" title="E18">A7</font></span>: 
<font color="Maroon" title="c2">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c3">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T23" onmouseover="rs('yellow_0/T23')" onmouseout="rh()">YELLOW_0:23</a></span>;<br/>
<a NAME="E6:28_2"/>
<font color="Maroon" title="c2">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c3">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c3">b</font>
 
;<br/>
<a NAME="E7:28_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c3">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="yellow11.html#K3" title="YELLOW11:func.3">[#</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c3">b</font></span>)</span>,<font color="Maroon" title="c2">a</font></span><a href="yellow11.html#K3" title="YELLOW11:func.3">#]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:28_2"><span class="lab"><font color="Green" title="E18">A7</font></span></a>, <a class="ref" href="yellow11.html#D4" target="_self" onmouseover="rs('yellow11/D4')" onmouseout="rh()">Def4</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">s2</font> =  <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p1"><a href="yellow11.html#K3" title="YELLOW11:func.3">[#</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c3">b</font></span>)</span>,<font color="Maroon" title="c2">a</font></span><a href="yellow11.html#K3" title="YELLOW11:func.3">#]</a></span> as   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="yellow_0.html#V4" title="YELLOW_0:attr.4">full</a>  <a href="lattice5.html#NM2" title="LATTICE5:NM.2">Sublattice</a> of <font color="Maroon" title="c1">L</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#D15" onmouseover="rs('yellow_0/D15')" onmouseout="rh()">YELLOW_0:def 15</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">f1</font> = <font color="Maroon" title="c5">f</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c6">s1</font>,<font color="Maroon" title="c7">s2</font> ;<br/>
<a NAME="E10:28_2"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2"><a href="yellow11.html#K3" title="YELLOW11:func.3">[#</a><span class="default"><font color="Maroon" title="c3">b</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">b</font></span>)</span></span><a href="yellow11.html#K3" title="YELLOW11:func.3">#]</a></span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E11:28_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A8</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="yellow11.html#K3" title="YELLOW11:func.3">[#</a><span class="default"><font color="Maroon" title="c3">b</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">b</font></span>)</span></span><a href="yellow11.html#K3" title="YELLOW11:func.3">#]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#D15" onmouseover="rs('yellow_0/D15')" onmouseout="rh()">YELLOW_0:def 15</a></span>;<br/>
<a NAME="E12:28_2"/>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2"><a href="yellow11.html#K3" title="YELLOW11:func.3">[#</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c3">b</font></span>)</span>,<font color="Maroon" title="c2">a</font></span><a href="yellow11.html#K3" title="YELLOW11:func.3">#]</a></span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c8">f1</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="28_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2"><a href="yellow11.html#K3" title="YELLOW11:func.3">[#</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c3">b</font></span>)</span>,<font color="Maroon" title="c2">a</font></span><a href="yellow11.html#K3" title="YELLOW11:func.3">#]</a></span></span>)</span> or <font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c8">f1</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:28_2_1"/>
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2"><a href="yellow11.html#K3" title="YELLOW11:func.3">[#</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c3">b</font></span>)</span>,<font color="Maroon" title="c2">a</font></span><a href="yellow11.html#K3" title="YELLOW11:func.3">#]</a></span></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c8">f1</font></span><br/>

<a NAME="E2:28_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A9</font></span>: 
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="yellow11.html#K3" title="YELLOW11:func.3">[#</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c3">b</font></span>)</span>,<font color="Maroon" title="c2">a</font></span><a href="yellow11.html#K3" title="YELLOW11:func.3">#]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#D15" onmouseover="rs('yellow_0/D15')" onmouseout="rh()">YELLOW_0:def 15</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">Y</font> = <font color="Maroon" title="c9">y</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="E4:28_2_1"/><span class="lab"><font color="Green" title="E21">A10</font></span>: 
<font color="Maroon" title="c2">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c3">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c10">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:28_2_1"><span class="lab"><font color="Green" title="E20">A9</font></span></a>, <a class="ref" href="yellow11.html#D4" target="_self" onmouseover="rs('yellow11/D4')" onmouseout="rh()">Def4</a></span>;<br/>
<a NAME="E5:28_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c10">Y</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_1.html#T2" onmouseover="rs('waybel_1/T2')" onmouseout="rh()">WAYBEL_1:2</a></span>;<br/>
<a NAME="E6:28_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A11</font></span>: 
<font color="Maroon" title="c3">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c10">Y</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#T17" onmouseover="rs('lattice3/T17')" onmouseout="rh()">LATTICE3:17</a></span>;<br/>
<a NAME="E7:28_2_1"/><span class="lab"><font color="Green" title="E23">A12</font></span>: 
<font color="Maroon" title="c10">Y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:28_2_1"><span class="lab"><font color="Green" title="E20">A9</font></span></a>, <a class="ref" href="yellow11.html#D4" target="_self" onmouseover="rs('yellow11/D4')" onmouseout="rh()">Def4</a></span>;<br/>
<a NAME="E8:28_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">Y</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_1.html#T2" onmouseover="rs('waybel_1/T2')" onmouseout="rh()">WAYBEL_1:2</a></span>;<br/>
<a NAME="E9:28_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A13</font></span>: 
<font color="Maroon" title="c10">Y</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="yellow11.html#K3" title="YELLOW11:func.3">[#</a><span class="default"><font color="Maroon" title="c3">b</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">b</font></span>)</span></span><a href="yellow11.html#K3" title="YELLOW11:func.3">#]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:28_2_1"><span class="lab"><font color="Green" title="E22">A11</font></span></a>, <a class="ref" href="yellow11.html#D4" target="_self" onmouseover="rs('yellow11/D4')" onmouseout="rh()">Def4</a></span>;<br/>
<a NAME="E10:28_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A14</font></span>: 
<font color="Maroon" title="c10">Y</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2"><a href="yellow11.html#K3" title="YELLOW11:func.3">[#</a><span class="default"><font color="Maroon" title="c3">b</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">b</font></span>)</span></span><a href="yellow11.html#K3" title="YELLOW11:func.3">#]</a></span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#D15" onmouseover="rs('yellow_0/D15')" onmouseout="rh()">YELLOW_0:def 15</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">f1yb</font> = <font color="Maroon" title="c8">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">Y</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">b</font></span>)</span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:28"><span class="lab"><font color="Green" title="E16">A5</font></span></a></span>;<br/>
<font color="Maroon" title="c11">f1yb</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">Y</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c2">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:28"><span class="lab"><font color="Green" title="E16">A5</font></span></a>, <a class="txt" href="#E10:28_2_1"><span class="lab"><font color="Green" title="E25">A14</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c10">Y</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c2">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="txt" href="#E7:28_2_1"><span class="lab"><font color="Green" title="E23">A12</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c10">Y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:28_2_1"><span class="lab"><font color="Green" title="E21">A10</font></span></a>, <a class="ref" href="yellow_5.html#T8" onmouseover="rs('yellow_5/T8')" onmouseout="rh()">YELLOW_5:8</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E13:28_2_1"/>
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c8">f1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:28_2"><span class="lab"><font color="Green" title="E19">A8</font></span></a>, <a class="txt" href="#E9:28_2_1"><span class="lab"><font color="Green" title="E24">A13</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E13:28_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A15</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c8">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2"><a href="yellow11.html#K3" title="YELLOW11:func.3">[#</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c3">b</font></span>)</span>,<font color="Maroon" title="c2">a</font></span><a href="yellow11.html#K3" title="YELLOW11:func.3">#]</a></span></span>)</span>
 
;<br/>
<a NAME="E14:28_2"/><span class="lab"><font color="Green" title="E21">A16</font></span>: 
 for <font color="Olive" title="b1">x</font>, <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="c6">s1</font> holds <br/> ( <font color="Olive" title="b1">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b2">y</font> iff <font color="Maroon" title="c8">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c8">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">y</font> )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="28_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">x</font>, <font color="Maroon" title="c10">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c6">s1</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c10">y</font> iff <font color="Maroon" title="c8">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c8">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">y</font> )</span><br/>

<a NAME="E1:28_2_2"/><span class="lab"><font color="Green" title="E22">A17</font></span>: 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="yellow11.html#K3" title="YELLOW11:func.3">[#</a><span class="default"><font color="Maroon" title="c3">b</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">b</font></span>)</span></span><a href="yellow11.html#K3" title="YELLOW11:func.3">#]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#D15" onmouseover="rs('yellow_0/D15')" onmouseout="rh()">YELLOW_0:def 15</a></span>;<br/>
<a NAME="E2:28_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="yellow11.html#K3" title="YELLOW11:func.3">[#</a><span class="default"><font color="Maroon" title="c3">b</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">b</font></span>)</span></span><a href="yellow11.html#K3" title="YELLOW11:func.3">#]</a></span>
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">X</font> = <font color="Maroon" title="c9">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="E4:28_2_2"/>
<font color="Maroon" title="c10">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="yellow11.html#K3" title="YELLOW11:func.3">[#</a><span class="default"><font color="Maroon" title="c3">b</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">b</font></span>)</span></span><a href="yellow11.html#K3" title="YELLOW11:func.3">#]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28_2_2"><span class="lab"><font color="Green" title="E22">A17</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">Y</font> = <font color="Maroon" title="c10">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">f1Y</font> = <font color="Maroon" title="c8">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">Y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:28"><span class="lab"><font color="Green" title="E16">A5</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">f1X</font> = <font color="Maroon" title="c8">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:28"><span class="lab"><font color="Green" title="E16">A5</font></span></a></span>;<br/>
<span class="kw">thus </span><a NAME="E8:28_2_2"/>
( <font color="Maroon" title="c9">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c10">y</font> implies <font color="Maroon" title="c8">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c8">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">y</font> )
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c8">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">y</font> implies <font color="Maroon" title="c9">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c10">y</font> )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="28_2_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:28_2_2_1"/>
<font color="Maroon" title="c9">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c10">y</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c8">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">y</font></span><br/>

<a NAME="E2:28_2_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">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="c9">x</font>,<font color="Maroon" title="c10">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c6">s1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="orders_2.html#D5" onmouseover="rs('orders_2/D5')" onmouseout="rh()">ORDERS_2:def 5</a></span>;<br/>
<a NAME="E3:28_2_2_1"/>
 the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c6">s1</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="ref" href="yellow_0.html#D13" onmouseover="rs('yellow_0/D13')" onmouseout="rh()">YELLOW_0:def 13</a></span>;<br/>
<a NAME="E4:28_2_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A19</font></span>: 
<font color="Maroon" title="c11">X</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="#E2:28_2_2_1"><span class="lab"><font color="Green" title="E23">A18</font></span></a>, <a class="ref" href="orders_2.html#D5" onmouseover="rs('orders_2/D5')" onmouseout="rh()">ORDERS_2:def 5</a></span>;<br/>
<a NAME="E5:28_2_2_1"/><span class="lab"><font color="Green" title="E25">A20</font></span>: 
<font color="Maroon" title="c13">f1Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">Y</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c2">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:28"><span class="lab"><font color="Green" title="E16">A5</font></span></a></span>;<br/>
<a NAME="E6:28_2_2_1"/>
<font color="Maroon" title="c14">f1X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">X</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c2">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:28"><span class="lab"><font color="Green" title="E16">A5</font></span></a></span>;<br/>
<a NAME="E7:28_2_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">f1X</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c13">f1Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:28_2_2_1"><span class="lab"><font color="Green" title="E24">A19</font></span></a>, <a class="txt" href="#E5:28_2_2_1"><span class="lab"><font color="Green" title="E25">A20</font></span></a>, <a class="ref" href="waybel_1.html#T1" onmouseover="rs('waybel_1/T1')" onmouseout="rh()">WAYBEL_1:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:28_2_2_1"/>
<font color="Maroon" title="c8">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c8">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T60" onmouseover="rs('yellow_0/T60')" onmouseout="rh()">YELLOW_0:60</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E9:28_2_2"/>
( <font color="Maroon" title="c8">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c8">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">y</font> implies <font color="Maroon" title="c9">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c10">y</font> )
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="28_2_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:28_2_2_2"/>
<font color="Maroon" title="c8">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c8">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">y</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c10">y</font></span><br/>

<a NAME="E2:28_2_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A21</font></span>: 
<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="c8">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">y</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c7">s2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="orders_2.html#D5" onmouseover="rs('orders_2/D5')" onmouseout="rh()">ORDERS_2:def 5</a></span>;<br/>
<a NAME="E3:28_2_2_2"/>
 the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c7">s2</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="ref" href="yellow_0.html#D13" onmouseover="rs('yellow_0/D13')" onmouseout="rh()">YELLOW_0:def 13</a></span>;<br/>
<a NAME="E4:28_2_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A22</font></span>: 
<font color="Maroon" title="c14">f1X</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c13">f1Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:28_2_2_2"><span class="lab"><font color="Green" title="E23">A21</font></span></a>, <a class="ref" href="orders_2.html#D5" onmouseover="rs('orders_2/D5')" onmouseout="rh()">ORDERS_2:def 5</a></span>;<br/>
<a NAME="E5:28_2_2_2"/><span class="lab"><font color="Green" title="E25">A23</font></span>: 
<font color="Maroon" title="c13">f1Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">Y</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c2">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:28"><span class="lab"><font color="Green" title="E16">A5</font></span></a></span>;<br/>
<a NAME="E6:28_2_2_2"/><span class="lab"><font color="Green" title="E26">A24</font></span>: 
<font color="Maroon" title="c3">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c11">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28_2_2"><span class="lab"><font color="Green" title="E22">A17</font></span></a>, <a class="ref" href="yellow11.html#D4" target="_self" onmouseover="rs('yellow11/D4')" onmouseout="rh()">Def4</a></span>;<br/>
<a NAME="E7:28_2_2_2"/>
<font color="Maroon" title="c14">f1X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">X</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c2">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:28"><span class="lab"><font color="Green" title="E16">A5</font></span></a></span>;<br/>
<a NAME="E8:28_2_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c11">X</font></span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c12">Y</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:28_2_2_2"><span class="lab"><font color="Green" title="E24">A22</font></span></a>, <a class="txt" href="#E5:28_2_2_2"><span class="lab"><font color="Green" title="E25">A23</font></span></a>, <a class="ref" href="waybel_1.html#T2" onmouseover="rs('waybel_1/T2')" onmouseout="rh()">WAYBEL_1:2</a></span>;<br/>
<a NAME="E9:28_2_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A25</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c11">X</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c12">Y</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="txt" href="#E6:28_2_2_2"><span class="lab"><font color="Green" title="E26">A24</font></span></a></span>;<br/>
<a NAME="E10:28_2_2_2"/><span class="lab"><font color="Green" title="E28">A26</font></span>: 
<font color="Maroon" title="c11">X</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c2">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28_2_2"><span class="lab"><font color="Green" title="E22">A17</font></span></a>, <a class="ref" href="yellow11.html#D4" target="_self" onmouseover="rs('yellow11/D4')" onmouseout="rh()">Def4</a></span>;<br/>
<a NAME="E11:28_2_2_2"/>
<font color="Maroon" title="c3">b</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="#E1:28_2_2"><span class="lab"><font color="Green" title="E22">A17</font></span></a>, <a class="ref" href="yellow11.html#D4" target="_self" onmouseover="rs('yellow11/D4')" onmouseout="rh()">Def4</a></span>;<br/>
<a NAME="E12:28_2_2_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c11">X</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c12">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="txt" href="#E9:28_2_2_2"><span class="lab"><font color="Green" title="E27">A25</font></span></a></span>;<br/>
<a NAME="E13:28_2_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A27</font></span>: 
<font color="Maroon" title="c11">X</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c12">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:28_2_2_2"><span class="lab"><font color="Green" title="E28">A26</font></span></a>, <a class="ref" href="yellow_5.html#T10" onmouseover="rs('yellow_5/T10')" onmouseout="rh()">YELLOW_5:10</a></span>;<br/>
<a NAME="E14:28_2_2_2"/>
<font color="Maroon" title="c12">Y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c2">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28_2_2"><span class="lab"><font color="Green" title="E22">A17</font></span></a>, <a class="ref" href="yellow11.html#D4" target="_self" onmouseover="rs('yellow11/D4')" onmouseout="rh()">Def4</a></span>;<br/>
<a NAME="E15:28_2_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c11">X</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="#E13:28_2_2_2"><span class="lab"><font color="Green" title="E29">A27</font></span></a>, <a class="ref" href="yellow_5.html#T10" onmouseover="rs('yellow_5/T10')" onmouseout="rh()">YELLOW_5:10</a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:28_2_2_2"/>
<font color="Maroon" title="c9">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c10">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T60" onmouseover="rs('yellow_0/T60')" onmouseout="rh()">YELLOW_0:60</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>


</div><span class="kw">end;</span></div>
<a NAME="E15:28_2"/>
<font color="Maroon" title="c8">f1</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">V13</a>()
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="28_2_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">x1</font>, <font color="Maroon" title="c10">x2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="funct_1.html#D4" onmouseover="rs('funct_1/D4')" onmouseout="rh()">FUNCT_1:def 4</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c9">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="xtuple_0.html#K9" title="XTUPLE_0:func.9">K21</a>(<font color="Maroon" title="c8">f1</font>) or  not <font color="Maroon" title="c10">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="xtuple_0.html#K9" title="XTUPLE_0:func.9">K21</a>(<font color="Maroon" title="c8">f1</font>) or  not <font color="Maroon" title="c8">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x2</font> or <font color="Maroon" title="c9">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x2</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:28_2_3"/><span class="lab"><font color="Green" title="E22">A28</font></span>: 
<font color="Maroon" title="c9">x1</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="c8">f1</font>
 <span class="kw">and </span><br/><a NAME="E2:28_2_3"/><span class="lab"><font color="Green" title="E23">A29</font></span>: 
<font color="Maroon" title="c10">x2</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="c8">f1</font>
 <span class="kw">and </span><br/><a NAME="E3:28_2_3"/><span class="lab"><font color="Green" title="E24">A30</font></span>: 
<font color="Maroon" title="c8">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x2</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c11">X2</font> = <font color="Maroon" title="c10">x2</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:28_2"><span class="lab"><font color="Green" title="E19">A8</font></span></a>, <a class="txt" href="#E2:28_2_3"><span class="lab"><font color="Green" title="E23">A29</font></span></a></span>;<br/>
<a NAME="E5:28_2_3"/><span class="lab"><font color="Green" title="E25">A31</font></span>: 
<font color="Maroon" title="c3">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c11">X2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:28_2"><span class="lab"><font color="Green" title="E19">A8</font></span></a>, <a class="txt" href="#E2:28_2_3"><span class="lab"><font color="Green" title="E23">A29</font></span></a>, <a class="ref" href="yellow11.html#D4" target="_self" onmouseover="rs('yellow11/D4')" onmouseout="rh()">Def4</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">X1</font> = <font color="Maroon" title="c9">x1</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:28_2"><span class="lab"><font color="Green" title="E19">A8</font></span></a>, <a class="txt" href="#E1:28_2_3"><span class="lab"><font color="Green" title="E22">A28</font></span></a></span>;<br/>
<a NAME="E7:28_2_3"/><span class="lab"><font color="Green" title="E26">A32</font></span>: 
<font color="Maroon" title="c3">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c12">X1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:28_2"><span class="lab"><font color="Green" title="E19">A8</font></span></a>, <a class="txt" href="#E1:28_2_3"><span class="lab"><font color="Green" title="E22">A28</font></span></a>, <a class="ref" href="yellow11.html#D4" target="_self" onmouseover="rs('yellow11/D4')" onmouseout="rh()">Def4</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">f1X1</font> = <font color="Maroon" title="c8">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">X1</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:28"><span class="lab"><font color="Green" title="E16">A5</font></span></a>, <a class="txt" href="#E1:28_2_3"><span class="lab"><font color="Green" title="E22">A28</font></span></a></span>;<br/>
<a NAME="E9:28_2_3"/><span class="lab"><font color="Green" title="E27">A33</font></span>: 
<font color="Maroon" title="c13">f1X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">X1</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c2">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:28"><span class="lab"><font color="Green" title="E16">A5</font></span></a>, <a class="txt" href="#E1:28_2_3"><span class="lab"><font color="Green" title="E22">A28</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">f1X2</font> = <font color="Maroon" title="c8">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">X2</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:28"><span class="lab"><font color="Green" title="E16">A5</font></span></a>, <a class="txt" href="#E2:28_2_3"><span class="lab"><font color="Green" title="E23">A29</font></span></a></span>;<br/>
<a NAME="E11:28_2_3"/><span class="lab"><font color="Green" title="E28">A34</font></span>: 
<font color="Maroon" title="c14">f1X2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">X2</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c2">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:28"><span class="lab"><font color="Green" title="E16">A5</font></span></a>, <a class="txt" href="#E2:28_2_3"><span class="lab"><font color="Green" title="E23">A29</font></span></a></span>;<br/>
<a NAME="E12:28_2_3"/><span class="lab"><font color="Green" title="E29">A35</font></span>: 
<font color="Maroon" title="c11">X2</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:28_2"><span class="lab"><font color="Green" title="E19">A8</font></span></a>, <a class="txt" href="#E2:28_2_3"><span class="lab"><font color="Green" title="E23">A29</font></span></a>, <a class="ref" href="yellow11.html#D4" target="_self" onmouseover="rs('yellow11/D4')" onmouseout="rh()">Def4</a></span>;<br/>
<a NAME="E13:28_2_3"/>
<font color="Maroon" title="c12">X1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:28_2"><span class="lab"><font color="Green" title="E19">A8</font></span></a>, <a class="txt" href="#E1:28_2_3"><span class="lab"><font color="Green" title="E22">A28</font></span></a>, <a class="ref" href="yellow11.html#D4" target="_self" onmouseover="rs('yellow11/D4')" onmouseout="rh()">Def4</a></span>;<br/>
<span class="kw">then </span><font color="Maroon" title="c12">X1</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c12">X1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_5.html#T10" onmouseover="rs('yellow_5/T10')" onmouseout="rh()">YELLOW_5:10</a></span>
<br/>.= 
<font color="Maroon" title="c3">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c11">X2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="txt" href="#E3:28_2_3"><span class="lab"><font color="Green" title="E24">A30</font></span></a>, <a class="txt" href="#E7:28_2_3"><span class="lab"><font color="Green" title="E26">A32</font></span></a>, <a class="txt" href="#E9:28_2_3"><span class="lab"><font color="Green" title="E27">A33</font></span></a>, <a class="txt" href="#E11:28_2_3"><span class="lab"><font color="Green" title="E28">A34</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c11">X2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="txt" href="#E5:28_2_3"><span class="lab"><font color="Green" title="E25">A31</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c11">X2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:28_2_3"><span class="lab"><font color="Green" title="E29">A35</font></span></a>, <a class="ref" href="yellow_5.html#T10" onmouseover="rs('yellow_5/T10')" onmouseout="rh()">YELLOW_5:10</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E15:28_2_3"/>
<font color="Maroon" title="c9">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E16:28_2"/>
<font color="Maroon" title="c5">f</font> is  <a href="waybel_0.html#V23" title="WAYBEL_0:attr.23">isomorphic</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:28_2"><span class="lab"><font color="Green" title="E20">A15</font></span></a>, <a class="txt" href="#E14:28_2"><span class="lab"><font color="Green" title="E21">A16</font></span></a>, <a class="ref" href="waybel_0.html#T66" onmouseover="rs('waybel_0/T66')" onmouseout="rh()">WAYBEL_0:66</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>


</div>
