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<b>let </b><font color="Maroon" title="c1">L1</font>, <font color="Maroon" title="c2">L2</font> be  <a href="yellow_0.html#V1" title="YELLOW_0:attr.1">lower-bounded</a> <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font> <a href="waybel_1.html#R5" title="WAYBEL_1:pred.5">are_isomorphic</a>  &amp; <font color="Maroon" title="c1">L1</font> is <a href="yellow11.html#V1" title="YELLOW11:attr.1">modular</a> implies <font color="Maroon" title="c2">L2</font> is <a href="yellow11.html#V1" title="YELLOW11:attr.1">modular</a> )</span><br/>

<b>assume </b><a NAME="E1:11"/><i><font color="Green" title="E8">A1</font></i>: 
( <font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font> <a href="waybel_1.html#R5" title="WAYBEL_1:pred.5">are_isomorphic</a>  &amp; <font color="Maroon" title="c1">L1</font> is <a href="yellow11.html#V1" title="YELLOW11:attr.1">modular</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">L2</font> is <a href="yellow11.html#V1" title="YELLOW11:attr.1">modular</a></span><br/>

<b>then consider </b><font color="Maroon" title="c3">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font><b> such that </b><br/><a NAME="E3:11"/><i><font color="Green" title="E9">A2</font></i>: 
<font color="Maroon" title="c3">f</font> is <a href="waybel_0.html#V23" title="WAYBEL_0:attr.23">isomorphic</a>
 <b>by </b><i><a class="ref" href="waybel_1.html#D8" title="WAYBEL_1:def.8">WAYBEL_1:def 8</a></i>;<br/>
<a NAME="E4:11"/>
( <font color="Maroon" title="c3">f</font> is <a href="waybel_0.html#V17" title="WAYBEL_0:attr.17">infs-preserving</a> &amp; <font color="Maroon" title="c3">f</font> is <a href="waybel_0.html#V18" title="WAYBEL_0:attr.18">sups-preserving</a> )
 
<b>by </b><i><a class="txt" href="lattice8.html#E3:11"><i><font color="Green" title="E9">A2</font></i></a>, <a class="ref" href="waybel13.html#T20" title="WAYBEL13:th.20">WAYBEL13:20</a></i>;<br/>
<a NAME="E5:11"/><b>then </b><i><font color="Green" title="E10">A3</font></i>: 
( <font color="Maroon" title="c3">f</font> is <a href="waybel_0.html#V19" title="WAYBEL_0:attr.19">meet-preserving</a> &amp; <font color="Maroon" title="c3">f</font> is <a href="waybel_0.html#V20" title="WAYBEL_0:attr.20">join-preserving</a> )
 
<b>by </b><i><a class="ref" href="lattice8.html#T4" target="_self" title="LATTICE8:th.4">Th4</a></i>;<br/>
<b>let </b><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</font>, <font color="Maroon" title="c6">c</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L2</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow11.html#D3" title="YELLOW11:def.3">YELLOW11:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c4">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">c</font> or <font color="Maroon" title="c4">a</font> <a href="lattice3.html#K10" title="LATTICE3:func.10">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Maroon" title="c6">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="lattice3.html#K10" title="LATTICE3:func.10">"\/"</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Maroon" title="c6">c</font> )</span><br/>

<b>assume </b><a NAME="E6:11"/><i><font color="Green" title="E11">A4</font></i>: 
<font color="Maroon" title="c4">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">c</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">a</font> <a href="lattice3.html#K10" title="LATTICE3:func.10">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Maroon" title="c6">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="lattice3.html#K10" title="LATTICE3:func.10">"\/"</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Maroon" title="c6">c</font></span><br/>

<b>set </b><font color="Maroon" title="c7">A</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">a</font>;<br/>
<b>set </b><font color="Maroon" title="c8">B</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">b</font>;<br/>
<b>set </b><font color="Maroon" title="c9">C</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">c</font>;<br/>
<a NAME="E7:11"/><i><font color="Green" title="E12">A5</font></i>: 
( <font color="Maroon" title="c3">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">f</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 <font color="Maroon" title="c2">L2</font> )
 
<b>by </b><i><a class="txt" href="lattice8.html#E3:11"><i><font color="Green" title="E9">A2</font></i></a>, <a class="ref" href="waybel_0.html#T66" title="WAYBEL_0:th.66">WAYBEL_0:66</a></i>;<br/>
<a NAME="E8:11"/><b>then </b><i><font color="Green" title="E13">A6</font></i>: 
( <font color="Maroon" title="c3">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp; <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">f</font> &amp; <font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">f</font> &amp; <font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">f</font> &amp; <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">a</font></span>)</span> &amp; <font color="Maroon" title="c5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">b</font></span>)</span> &amp; <font color="Maroon" title="c6">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">c</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="funct_1.html#T57" title="FUNCT_1:th.57">FUNCT_1:57</a></i>;<br/>
<a NAME="E9:11"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> )
 
<b>by </b><i><a class="txt" href="lattice8.html#E7:11"><i><font color="Green" title="E12">A5</font></i></a>, <a class="ref" href="funct_1.html#T54" title="FUNCT_1:th.54">FUNCT_1:54</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c10">A</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c11">B</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">b</font>, <font color="Maroon" title="c12">C</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">c</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font> ;<br/>
<a NAME="E11:11"/>
<font color="Maroon" title="c10">A</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c12">C</font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E3:11"><i><font color="Green" title="E9">A2</font></i></a>, <a class="txt" href="lattice8.html#E6:11"><i><font color="Green" title="E11">A4</font></i></a>, <a class="txt" href="lattice8.html#E8:11"><i><font color="Green" title="E13">A6</font></i></a>, <a class="ref" href="waybel_0.html#T66" title="WAYBEL_0:th.66">WAYBEL_0:66</a></i>;<br/>
<a NAME="E12:11"/><b>then </b><i><font color="Green" title="E14">A7</font></i>: 
<font color="Maroon" title="c10">A</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">B</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c12">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">A</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">B</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c12">C</font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E1:11"><i><font color="Green" title="E8">A1</font></i></a>, <a class="ref" href="yellow11.html#D3" title="YELLOW11:def.3">YELLOW11:def 3</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c4">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c6">c</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">A</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">B</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c12">C</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="lattice8.html#E5:11"><i><font color="Green" title="E10">A3</font></i></a>, <a class="txt" href="lattice8.html#E8:11"><i><font color="Green" title="E13">A6</font></i></a>, <a class="ref" href="waybel_6.html#T1" title="WAYBEL_6:th.1">WAYBEL_6:1</a></i>
<br/>.= 
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">A</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">B</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c12">C</font></span>)</span>
<b>by </b><i><a class="txt" href="lattice8.html#E5:11"><i><font color="Green" title="E10">A3</font></i></a>, <a class="txt" href="lattice8.html#E12:11"><i><font color="Green" title="E14">A7</font></i></a>, <a class="ref" href="waybel_6.html#T2" title="WAYBEL_6:th.2">WAYBEL_6:2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">A</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">B</font></span>)</span></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">C</font></span>)</span>
<b>by </b><i><a class="txt" href="lattice8.html#E5:11"><i><font color="Green" title="E10">A3</font></i></a>, <a class="ref" href="waybel_6.html#T1" title="WAYBEL_6:th.1">WAYBEL_6:1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c6">c</font>
<b>by </b><i><a class="txt" href="lattice8.html#E5:11"><i><font color="Green" title="E10">A3</font></i></a>, <a class="txt" href="lattice8.html#E8:11"><i><font color="Green" title="E13">A6</font></i></a>, <a class="ref" href="waybel_6.html#T2" title="WAYBEL_6:th.2">WAYBEL_6:2</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
