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<span class="kw">let </span><font color="Maroon" title="c1">S</font>, <font color="Maroon" title="c2">T</font> be    <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font> <a href="waybel_1.html#R5" title="WAYBEL_1:pred.5">are_isomorphic</a>  &amp; <font color="Maroon" title="c1">S</font> is  <a href="yellow_0.html#V2" title="YELLOW_0:attr.2">upper-bounded</a>  implies <font color="Maroon" title="c2">T</font> is  <a href="yellow_0.html#V2" title="YELLOW_0:attr.2">upper-bounded</a>  )</span><br/>

<span class="kw">given </span><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">S</font>,<font color="Maroon" title="c2">T</font><span class="kw"> such that </span><a NAME="E2:25"/><span class="lab"><font color="Green" title="E6">A1</font></span>: 
<font color="Maroon" title="c3">f</font> is  <a href="waybel_0.html#V23" title="WAYBEL_0:attr.23">isomorphic</a> 
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="waybel_1.html#D8" title="WAYBEL_1:def.8">WAYBEL_1:def 8</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="c1">S</font> is  <a href="yellow_0.html#V2" title="YELLOW_0:attr.2">upper-bounded</a>  or <font color="Maroon" title="c2">T</font> is  <a href="yellow_0.html#V2" title="YELLOW_0:attr.2">upper-bounded</a>  )</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:25_1"/>
( ( not <font color="Maroon" title="c1">S</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  &amp; not <font color="Maroon" title="c2">T</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  ) or <font color="Maroon" title="c1">S</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  or <font color="Maroon" title="c2">T</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1"><span class="kw">suppose </span></a><a NAME="E1:25_1_1"/>
( not <font color="Maroon" title="c1">S</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  &amp; not <font color="Maroon" title="c2">T</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</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="c1">S</font> is  <a href="yellow_0.html#V2" title="YELLOW_0:attr.2">upper-bounded</a>  or <font color="Maroon" title="c2">T</font> is  <a href="yellow_0.html#V2" title="YELLOW_0:attr.2">upper-bounded</a>  )</span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">s</font> = <font color="Maroon" title="c1">S</font>, <font color="Maroon" title="c5">t</font> = <font color="Maroon" title="c2">T</font> as  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  ;<br/><span class="kw">given </span><font color="Maroon" title="c6">X</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font><span class="kw"> such that </span><a NAME="E4:25_1_1"/><span class="lab"><font color="Green" title="E7">A2</font></span>: 
<font color="Maroon" title="c6">X</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="yellow_0.html#D5" title="YELLOW_0:def.5">YELLOW_0:def 5</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">T</font> is  <a href="yellow_0.html#V2" title="YELLOW_0:attr.2">upper-bounded</a> </span><br/><span class="kw">reconsider </span><font color="Maroon" title="c7">x</font> = <font color="Maroon" title="c6">X</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">s</font> ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c8">g</font> = <font color="Maroon" title="c3">f</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c5">t</font> ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c9">y</font> = <font color="Maroon" title="c8">g</font> <a href="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Maroon" title="c7">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">T</font> ;<br/><span class="kw">take </span>
<font color="Maroon" title="c9">y</font>
; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="yellow_0.html#D5" title="YELLOW_0:def.5">YELLOW_0:def 5</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Maroon" title="c9">y</font></span><br/><a NAME="E8:25_1_1"/>
<font color="Maroon" title="c9">y</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> <font color="Maroon" title="c8">g</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">S</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:25"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="txt" href="#E4:25_1_1"><span class="lab"><font color="Green" title="E7">A2</font></span></a>, <a class="ref" href="yellow_2.html#T16" title="YELLOW_2:th.16">YELLOW_2:16</a></span>;<br/><a NAME="E9:25_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">y</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c8">g</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T45" title="FUNCT_2:th.45">FUNCT_2:45</a></span>;<br/><span class="kw">hence </span><a NAME="E10:25_1_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Maroon" title="c9">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:25"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="ref" href="waybel_0.html#T66" title="WAYBEL_0:th.66">WAYBEL_0:66</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_2"><span class="kw">suppose </span></a><a NAME="E1:25_1_2"/>
( <font color="Maroon" title="c1">S</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  or <font color="Maroon" title="c2">T</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</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="c1">S</font> is  <a href="yellow_0.html#V2" title="YELLOW_0:attr.2">upper-bounded</a>  or <font color="Maroon" title="c2">T</font> is  <a href="yellow_0.html#V2" title="YELLOW_0:attr.2">upper-bounded</a>  )</span><br/><div class="add"><a NAME="E2:25_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">T</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:25"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="ref" href="waybel_0.html#D38" title="WAYBEL_0:def.38">WAYBEL_0:def 38</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">T</font> = <font color="Maroon" title="c2">T</font> as   <a href="yellow_0.html#V3" title="YELLOW_0:attr.3">bounded</a>   <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  ;<br/><a NAME="E4:25_1_2"/>
<font color="Maroon" title="c4">T</font> is  <a href="yellow_0.html#V1" title="YELLOW_0:attr.1">lower-bounded</a> 
 ;<br/><span class="kw">hence </span><a NAME="E5:25_1_2"/>
(  not <font color="Maroon" title="c1">S</font> is  <a href="yellow_0.html#V2" title="YELLOW_0:attr.2">upper-bounded</a>  or <font color="Maroon" title="c2">T</font> is  <a href="yellow_0.html#V2" title="YELLOW_0:attr.2">upper-bounded</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

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