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<b>let </b><font color="Maroon" title="c1">S</font> be  <a href="osalg_4.html#V2" title="OSALG_4:attr.2">locally_directed</a> <a href="osalg_1.html#NM3" title="OSALG_1:NM.3">OrderSortedSign</a>;<br/><b>let </b><font color="Maroon" title="c2">U1</font> be  <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a> <a href="osalg_1.html#NM5" title="OSALG_1:NM.5">OSAlgebra</a> of <font color="Maroon" title="c1">S</font>;<br/><b>let </b><font color="Maroon" title="c3">E</font> be  <a href="msualg_4.html#V3" title="MSUALG_4:attr.3">MSEquivalence-like</a>  <a href="osalg_4.html#M1" title="OSALG_4:mode.1">OrderSortedRelation</a> of <font color="Maroon" title="c2">U1</font>;<br/><b>let </b><font color="Maroon" title="c4">s</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font>;<br/><b>let </b><font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c6">y</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">U1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">s</font>;<br/>









<b>reconsider </b><font color="Maroon" title="c7">SU</font> = the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">U1</font> as   <a href="osalg_1.html#NM4" title="OSALG_1:NM.4">OrderSortedSet</a> of <font color="Maroon" title="c1">S</font> <b>by </b><i><a class="ref" href="osalg_1.html#T17" title="OSALG_1:th.17">OSALG_1:17</a></i>;<br/>
<a NAME="E2:40"/><i><font color="Green" title="E19">A1</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">U1</font> <a href="osalg_4.html#K4" title="OSALG_4:func.4">-carrier_of</a> <span class="p1">(<span class="default"><a href="osalg_4.html#K3" title="OSALG_4:func.3">CComp</a> <font color="Maroon" title="c4">s</font></span>)</span>
 
<b>by </b><i><a class="ref" href="osalg_4.html#T6" target="_self" title="OSALG_4:th.6">Th6</a></i>;<br/>
<a NAME="E3:40"/><i><font color="Green" title="E20">A2</font></i>: 
<font color="Maroon" title="c4">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="osalg_4.html#K3" title="OSALG_4:func.3">CComp</a> <font color="Maroon" title="c4">s</font>
 
<b>by </b><i><a class="ref" href="eqrel_1.html#T28" title="EQREL_1:th.28">EQREL_1:28</a></i>;<br/>
<a NAME="E4:40"/><i><font color="Green" title="E21">A3</font></i>: 
<font color="Maroon" title="c3">E</font> is <a href="osalg_4.html#V1" title="OSALG_4:attr.1">os-compatible</a>
 
<b>by </b><i><a class="ref" href="osalg_4.html#D3" target="_self" title="OSALG_4:def.3">Def3</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:40_1"/>
 <a href="osalg_4.html#K8" title="OSALG_4:func.8">OSClass</a> <font color="Maroon" title="c3">E</font>,<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="osalg_4.html#K8" title="OSALG_4:func.8">OSClass</a> <font color="Maroon" title="c3">E</font>,<font color="Maroon" title="c6">y</font>
 ;<br/><a NAME="E2:40_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="osalg_4.html#K5" title="OSALG_4:func.5">CompClass</a> <font color="Maroon" title="c3">E</font>,<span class="p1">(<span class="default"><a href="osalg_4.html#K3" title="OSALG_4:func.3">CComp</a> <font color="Maroon" title="c4">s</font></span>)</span>
 <b>by </b><i><a class="txt" href="osalg_4.html#E2:40"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="eqrel_1.html#T44" title="EQREL_1:th.44">EQREL_1:44</a></i>;<br/><b>then consider </b><font color="Maroon" title="c8">s1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font><b> such that </b><br/><a NAME="E4:40_1"/><i><font color="Green" title="E22">A4</font></i>: 
( <font color="Maroon" title="c8">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="osalg_4.html#K3" title="OSALG_4:func.3">CComp</a> <font color="Maroon" title="c4">s</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">E</font> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c8">s1</font> )
 <b>by </b><i><a class="ref" href="osalg_4.html#D11" target="_self" title="OSALG_4:def.11">Def11</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c9">sn1</font> = <font color="Maroon" title="c4">s</font>, <font color="Maroon" title="c10">s11</font> = <font color="Maroon" title="c8">s1</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> ;<br/><a NAME="E6:40_1"/><i><font color="Green" title="E23">A5</font></i>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">SU</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">s11</font> &amp; <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">SU</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">s11</font> )
 <b>by </b><i><a class="txt" href="osalg_4.html#E4:40_1"><i><font color="Green" title="E22">A4</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><b>consider </b><font color="Maroon" title="c11">s2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font><b> such that </b><br/><a NAME="E8:40_1"/><i><font color="Green" title="E24">A6</font></i>: 
( <font color="Maroon" title="c11">s2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="osalg_4.html#K3" title="OSALG_4:func.3">CComp</a> <font color="Maroon" title="c4">s</font> &amp; <font color="Maroon" title="c10">s11</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c11">s2</font> &amp; <font color="Maroon" title="c9">sn1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c11">s2</font> )
 <b>by </b><i><a class="txt" href="osalg_4.html#E3:40"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="osalg_4.html#E4:40_1"><i><font color="Green" title="E22">A4</font></i></a>, <a class="ref" href="waybel_0.html#D1" title="WAYBEL_0:def.1">WAYBEL_0:def 1</a></i>;<br/><a NAME="E9:40_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">E</font> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c11">s2</font>
 <b>by </b><i><a class="txt" href="osalg_4.html#E4:40"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="osalg_4.html#E4:40_1"><i><font color="Green" title="E22">A4</font></i></a>, <a class="txt" href="osalg_4.html#E6:40_1"><i><font color="Green" title="E23">A5</font></i></a>, <a class="txt" href="osalg_4.html#E8:40_1"><i><font color="Green" title="E24">A6</font></i></a>, <a class="ref" href="osalg_4.html#D1" target="_self" title="OSALG_4:def.1">Def1</a></i>;<br/><b>hence </b><a NAME="E10:40_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">E</font> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c4">s</font>
 <b>by </b><i><a class="txt" href="osalg_4.html#E4:40"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="osalg_4.html#E8:40_1"><i><font color="Green" title="E24">A6</font></i></a>, <a class="ref" href="osalg_4.html#D1" target="_self" title="OSALG_4:def.1">Def1</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E6:40"/><i><font color="Green" title="E22">A7</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">E</font> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c4">s</font>
 ;<br/>

<a NAME="E7:40"/><i><font color="Green" title="E23">A8</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">U1</font> <a href="osalg_4.html#K4" title="OSALG_4:func.4">-carrier_of</a> <span class="p1">(<span class="default"><a href="osalg_4.html#K3" title="OSALG_4:func.3">CComp</a> <font color="Maroon" title="c4">s</font></span>)</span>
 
<b>by </b><i><a class="ref" href="osalg_4.html#T6" target="_self" title="OSALG_4:th.6">Th6</a></i>;<br/>
<a NAME="E8:40"/>
<font color="Maroon" title="c4">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="osalg_4.html#K3" title="OSALG_4:func.3">CComp</a> <font color="Maroon" title="c4">s</font>
 
<b>by </b><i><a class="ref" href="eqrel_1.html#T28" title="EQREL_1:th.28">EQREL_1:28</a></i>;<br/>
<a NAME="E9:40"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="osalg_4.html#K5" title="OSALG_4:func.5">CompClass</a> <font color="Maroon" title="c3">E</font>,<span class="p1">(<span class="default"><a href="osalg_4.html#K3" title="OSALG_4:func.3">CComp</a> <font color="Maroon" title="c4">s</font></span>)</span>
 
<b>by </b><i><a class="txt" href="osalg_4.html#E6:40"><i><font color="Green" title="E22">A7</font></i></a>, <a class="ref" href="osalg_4.html#D11" target="_self" title="OSALG_4:def.11">Def11</a></i>;<br/>
<b>hence </b><a NAME="E10:40"/>
 <a href="osalg_4.html#K8" title="OSALG_4:func.8">OSClass</a> <font color="Maroon" title="c3">E</font>,<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="osalg_4.html#K8" title="OSALG_4:func.8">OSClass</a> <font color="Maroon" title="c3">E</font>,<font color="Maroon" title="c6">y</font>
 <b>by </b><i><a class="txt" href="osalg_4.html#E7:40"><i><font color="Green" title="E23">A8</font></i></a>, <a class="ref" href="eqrel_1.html#T44" title="EQREL_1:th.44">EQREL_1:44</a></i>;<br/>


</div>
