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<span class="kw">let </span><font color="Maroon" title="c1">S</font>, <font color="Maroon" title="c2">T</font> be  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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">f</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( ( <font color="Olive" title="b1">f</font> is    <a href="waybel_1.html#M1" title="WAYBEL_1:mode.1">Connection</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font> implies <font color="Olive" title="b1">f</font> is    <a href="waybel_1.html#M1" title="WAYBEL_1:mode.1">Connection</a> of <font color="Maroon" title="c1">S</font> <a href="lattice3.html#K7" title="LATTICE3:func.7">~</a> ,<font color="Maroon" title="c2">T</font> ) &amp; ( <font color="Olive" title="b1">f</font> is    <a href="waybel_1.html#M1" title="WAYBEL_1:mode.1">Connection</a> of <font color="Maroon" title="c1">S</font> <a href="lattice3.html#K7" title="LATTICE3:func.7">~</a> ,<font color="Maroon" title="c2">T</font> implies <font color="Olive" title="b1">f</font> is    <a href="waybel_1.html#M1" title="WAYBEL_1:mode.1">Connection</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font> ) &amp; ( <font color="Olive" title="b1">f</font> is    <a href="waybel_1.html#M1" title="WAYBEL_1:mode.1">Connection</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font> implies <font color="Olive" title="b1">f</font> is    <a href="waybel_1.html#M1" title="WAYBEL_1:mode.1">Connection</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font> <a href="lattice3.html#K7" title="LATTICE3:func.7">~</a>  ) &amp; ( <font color="Olive" title="b1">f</font> is    <a href="waybel_1.html#M1" title="WAYBEL_1:mode.1">Connection</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font> <a href="lattice3.html#K7" title="LATTICE3:func.7">~</a>  implies <font color="Olive" title="b1">f</font> is    <a href="waybel_1.html#M1" title="WAYBEL_1:mode.1">Connection</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font> ) &amp; ( <font color="Olive" title="b1">f</font> is    <a href="waybel_1.html#M1" title="WAYBEL_1:mode.1">Connection</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font> implies <font color="Olive" title="b1">f</font> is    <a href="waybel_1.html#M1" title="WAYBEL_1:mode.1">Connection</a> of <font color="Maroon" title="c1">S</font> <a href="lattice3.html#K7" title="LATTICE3:func.7">~</a> ,<font color="Maroon" title="c2">T</font> <a href="lattice3.html#K7" title="LATTICE3:func.7">~</a>  ) &amp; ( <font color="Olive" title="b1">f</font> is    <a href="waybel_1.html#M1" title="WAYBEL_1:mode.1">Connection</a> of <font color="Maroon" title="c1">S</font> <a href="lattice3.html#K7" title="LATTICE3:func.7">~</a> ,<font color="Maroon" title="c2">T</font> <a href="lattice3.html#K7" title="LATTICE3:func.7">~</a>  implies <font color="Olive" title="b1">f</font> is    <a href="waybel_1.html#M1" title="WAYBEL_1:mode.1">Connection</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font> ) )</span><br/>

<div><span class="lab"><font color="Green" title="E30">A1</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">S</font>, <font color="Maroon" title="c4">S1</font>, <font color="Maroon" title="c5">T</font>, <font color="Maroon" title="c6">T1</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"> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">S</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="c4">S1</font> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">T</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="c6">T1</font> implies for <font color="Olive" title="b1">a</font> being   <a href="waybel_1.html#M1" title="WAYBEL_1:mode.1">Connection</a> of <font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c5">T</font> holds  <font color="Olive" title="b1">a</font> is    <a href="waybel_1.html#M1" title="WAYBEL_1:mode.1">Connection</a> of <font color="Maroon" title="c4">S1</font>,<font color="Maroon" title="c6">T1</font> )</span><br/><span class="kw">assume </span><a NAME="E1:53_1"/><span class="lab"><font color="Green" title="E31">A2</font></span>: 
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">S</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="c4">S1</font> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">T</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="c6">T1</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">a</font> being   <a href="waybel_1.html#M1" title="WAYBEL_1:mode.1">Connection</a> of <font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c5">T</font> holds  <font color="Olive" title="b1">a</font> is    <a href="waybel_1.html#M1" title="WAYBEL_1:mode.1">Connection</a> of <font color="Maroon" title="c4">S1</font>,<font color="Maroon" title="c6">T1</font></span><br/><span class="kw">let </span><font color="Maroon" title="c7">a</font> be    <a href="waybel_1.html#M1" title="WAYBEL_1:mode.1">Connection</a> of <font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c5">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">a</font> is    <a href="waybel_1.html#M1" title="WAYBEL_1:mode.1">Connection</a> of <font color="Maroon" title="c4">S1</font>,<font color="Maroon" title="c6">T1</font></span><br/><span class="kw">consider </span><font color="Maroon" title="c8">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c5">T</font>, <font color="Maroon" title="c9">g</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c5">T</font>,<font color="Maroon" title="c3">S</font><span class="kw"> such that </span><br/><a NAME="E3:53_1"/><span class="lab"><font color="Green" title="E32">A3</font></span>: 
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="waybel_1.html#K1" title="WAYBEL_1:func.1">[</a><span class="default"><font color="Maroon" title="c8">f</font>,<font color="Maroon" title="c9">g</font></span><a href="waybel_1.html#K1" title="WAYBEL_1:func.1">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_1.html#D9" title="WAYBEL_1:def.9">WAYBEL_1:def 9</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c10">g</font> = <font color="Maroon" title="c9">g</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c6">T1</font>,<font color="Maroon" title="c4">S1</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:53_1"><span class="lab"><font color="Green" title="E31">A2</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c11">f</font> = <font color="Maroon" title="c8">f</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c4">S1</font>,<font color="Maroon" title="c6">T1</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:53_1"><span class="lab"><font color="Green" title="E31">A2</font></span></a></span>;<br/><a NAME="E6:53_1"/>
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="waybel_1.html#K1" title="WAYBEL_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">f</font>,<font color="Maroon" title="c10">g</font></span><a href="waybel_1.html#K1" title="WAYBEL_1:func.1">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:53_1"><span class="lab"><font color="Green" title="E32">A3</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E7:53_1"/>
<font color="Maroon" title="c7">a</font> is    <a href="waybel_1.html#M1" title="WAYBEL_1:mode.1">Connection</a> of <font color="Maroon" title="c4">S1</font>,<font color="Maroon" title="c6">T1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E2:53"/>
( <font color="Maroon" title="c1">S</font> <a href="lattice3.html#K7" title="LATTICE3:func.7">~</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</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="p1">(<span class="default">the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">S</font> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a> </span>)</span> #) &amp; <font color="Maroon" title="c2">T</font> <a href="lattice3.html#K7" title="LATTICE3:func.7">~</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T</font>,<span class="p1">(<span class="default">the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c2">T</font> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a> </span>)</span> #) )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:53"/>
for <font color="Olive" title="b1">f</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( ( <font color="Olive" title="b1">f</font> is    <a href="waybel_1.html#M1" title="WAYBEL_1:mode.1">Connection</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font> implies <font color="Olive" title="b1">f</font> is    <a href="waybel_1.html#M1" title="WAYBEL_1:mode.1">Connection</a> of <font color="Maroon" title="c1">S</font> <a href="lattice3.html#K7" title="LATTICE3:func.7">~</a> ,<font color="Maroon" title="c2">T</font> ) &amp; ( <font color="Olive" title="b1">f</font> is    <a href="waybel_1.html#M1" title="WAYBEL_1:mode.1">Connection</a> of <font color="Maroon" title="c1">S</font> <a href="lattice3.html#K7" title="LATTICE3:func.7">~</a> ,<font color="Maroon" title="c2">T</font> implies <font color="Olive" title="b1">f</font> is    <a href="waybel_1.html#M1" title="WAYBEL_1:mode.1">Connection</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font> ) &amp; ( <font color="Olive" title="b1">f</font> is    <a href="waybel_1.html#M1" title="WAYBEL_1:mode.1">Connection</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font> implies <font color="Olive" title="b1">f</font> is    <a href="waybel_1.html#M1" title="WAYBEL_1:mode.1">Connection</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font> <a href="lattice3.html#K7" title="LATTICE3:func.7">~</a>  ) &amp; ( <font color="Olive" title="b1">f</font> is    <a href="waybel_1.html#M1" title="WAYBEL_1:mode.1">Connection</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font> <a href="lattice3.html#K7" title="LATTICE3:func.7">~</a>  implies <font color="Olive" title="b1">f</font> is    <a href="waybel_1.html#M1" title="WAYBEL_1:mode.1">Connection</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font> ) &amp; ( <font color="Olive" title="b1">f</font> is    <a href="waybel_1.html#M1" title="WAYBEL_1:mode.1">Connection</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font> implies <font color="Olive" title="b1">f</font> is    <a href="waybel_1.html#M1" title="WAYBEL_1:mode.1">Connection</a> of <font color="Maroon" title="c1">S</font> <a href="lattice3.html#K7" title="LATTICE3:func.7">~</a> ,<font color="Maroon" title="c2">T</font> <a href="lattice3.html#K7" title="LATTICE3:func.7">~</a>  ) &amp; ( <font color="Olive" title="b1">f</font> is    <a href="waybel_1.html#M1" title="WAYBEL_1:mode.1">Connection</a> of <font color="Maroon" title="c1">S</font> <a href="lattice3.html#K7" title="LATTICE3:func.7">~</a> ,<font color="Maroon" title="c2">T</font> <a href="lattice3.html#K7" title="LATTICE3:func.7">~</a>  implies <font color="Olive" title="b1">f</font> is    <a href="waybel_1.html#M1" title="WAYBEL_1:mode.1">Connection</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:53"><span class="lab"><font color="Green" title="E30">A1</font></span></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/>


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