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<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:11"/><span class="lab"><font color="Green" title="E8">A1</font></span>: 
<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> 
 <span class="kw">and </span><br/><a NAME="E2:11"/><span class="lab"><font color="Green" title="E9">A2</font></span>: 
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<span class="kw">let </span><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font>, <font color="Maroon" title="c5">c</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L2</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="yellow11.html#D3" title="YELLOW11:def.3">YELLOW11:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c3">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">c</font> or <font color="Maroon" title="c3">a</font> <a href="lattice3.html#K10" title="LATTICE3:func.10">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Maroon" title="c5">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="lattice3.html#K10" title="LATTICE3:func.10">"\/"</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Maroon" title="c5">c</font> )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c6">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><span class="kw"> such that </span><br/><a NAME="E4:11"/><span class="lab"><font color="Green" title="E10">A3</font></span>: 
<font color="Maroon" title="c6">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="#E1:11"><span class="lab"><font color="Green" title="E8">A1</font></span></a>, <a class="ref" href="waybel_1.html#D8" title="WAYBEL_1:def.8">WAYBEL_1:def 8</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">C</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c6">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">c</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">A</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c6">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="c3">a</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">B</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c6">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">b</font>;<br/>
<a NAME="E5:11"/><span class="lab"><font color="Green" title="E11">A4</font></span>: 
( <font color="Maroon" title="c6">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">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> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:11"><span class="lab"><font color="Green" title="E10">A3</font></span></a>, <a class="ref" href="waybel_0.html#T66" title="WAYBEL_0:th.66">WAYBEL_0:66</a></span>;<br/>
<a NAME="E6:11"/><span class="kw">then </span><span class="lab"><font color="Green" title="E12">A5</font></span>: 
<font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">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="c6">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">b</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T57" title="FUNCT_1:th.57">FUNCT_1:57</a></span>;<br/>
<a NAME="E7:11"/><span class="lab"><font color="Green" title="E13">A6</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">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">c</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="c6">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:11"><span class="lab"><font color="Green" title="E11">A4</font></span></a>, <a class="ref" href="funct_1.html#T54" title="FUNCT_1:th.54">FUNCT_1:54</a></span>;<br/>
<a NAME="E8:11"/><span class="lab"><font color="Green" title="E14">A7</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c6">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="c3">a</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="c6">f</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">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">b</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="c6">f</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:11"><span class="lab"><font color="Green" title="E11">A4</font></span></a>, <a class="ref" href="funct_1.html#T54" title="FUNCT_1:th.54">FUNCT_1:54</a></span>;<br/>
<a NAME="E9:11"/><span class="lab"><font color="Green" title="E15">A8</font></span>: 
( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">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="c6">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="c3">a</font></span>)</span> &amp; <font color="Maroon" title="c5">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">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="c6">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">c</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:11"><span class="lab"><font color="Green" title="E11">A4</font></span></a>, <a class="ref" href="funct_1.html#T57" title="FUNCT_1:th.57">FUNCT_1:57</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">A</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c6">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="c3">a</font>, <font color="Maroon" title="c11">B</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c6">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">b</font>, <font color="Maroon" title="c12">C</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c6">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">c</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:11"><span class="lab"><font color="Green" title="E14">A7</font></span></a>, <a class="txt" href="#E7:11"><span class="lab"><font color="Green" title="E13">A6</font></span></a></span>;<br/>
<span class="kw">assume </span><a NAME="E11:11"/>
<font color="Maroon" title="c3">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">c</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">a</font> <a href="lattice3.html#K10" title="LATTICE3:func.10">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Maroon" title="c5">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="lattice3.html#K10" title="LATTICE3:func.10">"\/"</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Maroon" title="c5">c</font></span><br/>

<a NAME="E12:11"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:11"><span class="lab"><font color="Green" title="E10">A3</font></span></a>, <a class="txt" href="#E9:11"><span class="lab"><font color="Green" title="E15">A8</font></span></a>, <a class="ref" href="waybel_0.html#T66" title="WAYBEL_0:th.66">WAYBEL_0:66</a></span>;<br/>
<a NAME="E13:11"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A9</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:11"><span class="lab"><font color="Green" title="E9">A2</font></span></a>, <a class="ref" href="yellow11.html#D3" title="YELLOW11:def.3">YELLOW11:def 3</a></span>;<br/>
<a NAME="E14:11"/>
( <font color="Maroon" title="c6">f</font> is  <a href="waybel_0.html#V17" title="WAYBEL_0:attr.17">infs-preserving</a>  &amp; <font color="Maroon" title="c6">f</font> is  <a href="waybel_0.html#V18" title="WAYBEL_0:attr.18">sups-preserving</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:11"><span class="lab"><font color="Green" title="E10">A3</font></span></a>, <a class="ref" href="waybel13.html#T20" title="WAYBEL13:th.20">WAYBEL13:20</a></span>;<br/>
<a NAME="E15:11"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A10</font></span>: 
( <font color="Maroon" title="c6">f</font> is  <a href="waybel_0.html#V19" title="WAYBEL_0:attr.19">meet-preserving</a>  &amp; <font color="Maroon" title="c6">f</font> is  <a href="waybel_0.html#V20" title="WAYBEL_0:attr.20">join-preserving</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice8.html#T4" target="_self" title="LATTICE8:th.4">Th4</a></span>;<br/>
<span class="kw">hence </span><font color="Maroon" title="c3">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c5">c</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">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="c6">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:11"><span class="lab"><font color="Green" title="E12">A5</font></span></a>, <a class="txt" href="#E9:11"><span class="lab"><font color="Green" title="E15">A8</font></span></a>, <a class="ref" href="waybel_6.html#T1" title="WAYBEL_6:th.1">WAYBEL_6:1</a></span>
<br/>.= 
<font color="Maroon" title="c6">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:11"><span class="lab"><font color="Green" title="E17">A10</font></span></a>, <a class="txt" href="#E13:11"><span class="lab"><font color="Green" title="E16">A9</font></span></a>, <a class="ref" href="waybel_6.html#T2" title="WAYBEL_6:th.2">WAYBEL_6:2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">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="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">C</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:11"><span class="lab"><font color="Green" title="E17">A10</font></span></a>, <a class="ref" href="waybel_6.html#T1" title="WAYBEL_6:th.1">WAYBEL_6:1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c5">c</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:11"><span class="lab"><font color="Green" title="E17">A10</font></span></a>, <a class="txt" href="#E6:11"><span class="lab"><font color="Green" title="E12">A5</font></span></a>, <a class="txt" href="#E9:11"><span class="lab"><font color="Green" title="E15">A8</font></span></a>, <a class="ref" href="waybel_6.html#T2" title="WAYBEL_6:th.2">WAYBEL_6:2</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
