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<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="osalg_1.html#NM3">OrderSortedSign</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>set </b><font color="Maroon">c<sub>3</sub></font> = the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> as   <a href="struct_0.html#NM9">ManySortedSet</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>thus </b><a NAME="E2:58"/>
( <font color="Maroon">c<sub>2</sub></font> is <a href="osalg_1.html#V12" target="_self">order-sorted</a> implies the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> is   <a href="osalg_1.html#NM4">OrderSortedSet</a> of <font color="Maroon">c<sub>1</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:58_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="osalg_1.html#V12" target="_self">order-sorted</a>
 ;<br/>

<a NAME="E2:58_1"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="osalg_1.html#V11" target="_self">order-sorted</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="osalg_1.html#NM1">SortSymbol</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="osalg_1.html#D18" target="_self">OSALG_1:def 18</a><br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="osalg_1.html#NM1">SortSymbol</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>thus </b><a NAME="E1:58_1_1"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> implies <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="osalg_1.html#E1:58_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="osalg_1.html#D19" target="_self">Def19</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:58_1"/>
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> is   <a href="osalg_1.html#NM4">OrderSortedSet</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E3:58"/><i><font color="Green">E34</font></i>: 
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> is   <a href="osalg_1.html#NM4">OrderSortedSet</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="osalg_1.html#NM1">SortSymbol</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="osalg_1.html#D19" target="_self">OSALG_1:def 19</a><br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="osalg_1.html#NM1">SortSymbol</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>thus </b><a NAME="E4:58"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> implies the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="osalg_1.html#E3:58"><i><font color="Green">E34</font></i></a>, <a class="ref" href="osalg_1.html#D18" target="_self">Def18</a></i>;<br/>


</div>
