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<b>let </b><font color="Maroon">N1</font> be    <a href="msalimit.html#M2" target="_self">Binding</a> of <font color="Maroon">OAF</font>, <font color="Maroon">N2</font> be    <a href="msalimit.html#M2" target="_self">Binding</a> of <font color="Maroon">OAF</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:9_1_2"/><i><font color="Green">E97</font></i>: 
for <font color="Olive">i</font>, <font color="Olive">j</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">P</font>  st <font color="Olive">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">j</font> holds <br/><font color="Maroon">N1</font> <a href="binop_1.html#K1">.</a> <font color="Olive">j</font>,<font color="Olive">i</font> <a href="hidden.html#R1">=</a>  <a href="cqc_lang.html#K1">IFEQ</a> <font color="Olive">j</font>,<font color="Olive">i</font>,<span class="p1">(<span class="default"><a href="msualg_3.html#K2">id</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p2">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Olive">i</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msalimit.html#K1" target="_self">bind</a> <font color="Maroon">B</font>,<font color="Olive">i</font>,<font color="Olive">j</font></span>)</span> <a href="msualg_3.html#K3">**</a> <span class="p2">(<span class="default"><a href="msualg_3.html#K2">id</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p3">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Olive">i</font></span>)</span></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E2:9_1_2"/><i><font color="Green">E98</font></i>: 
for <font color="Olive">i</font>, <font color="Olive">j</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">P</font>  st <font color="Olive">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">j</font> holds <br/><font color="Maroon">N2</font> <a href="binop_1.html#K1">.</a> <font color="Olive">j</font>,<font color="Olive">i</font> <a href="hidden.html#R1">=</a>  <a href="cqc_lang.html#K1">IFEQ</a> <font color="Olive">j</font>,<font color="Olive">i</font>,<span class="p1">(<span class="default"><a href="msualg_3.html#K2">id</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p2">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Olive">i</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msalimit.html#K1" target="_self">bind</a> <font color="Maroon">B</font>,<font color="Olive">i</font>,<font color="Olive">j</font></span>)</span> <a href="msualg_3.html#K3">**</a> <span class="p2">(<span class="default"><a href="msualg_3.html#K2">id</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p3">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Olive">i</font></span>)</span></span>)</span></span>)</span>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">ij</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:9_1_2_1"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">ij</font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">P</font>
 ;<br/><b>then reconsider </b><font color="Maroon">i2</font> = <font color="Maroon">ij</font> <a href="mcart_1.html#K1">`1</a> , <font color="Maroon">i1</font> = <font color="Maroon">ij</font> <a href="mcart_1.html#K2">`2</a>  as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">P</font> <b>by </b><i><a class="ref" href="mcart_1.html#T10">MCART_1:10</a></i>;<br/><b>reconsider </b><font color="Maroon">i1</font> = <font color="Maroon">i1</font>, <font color="Maroon">i2</font> = <font color="Maroon">i2</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">P</font> ;<br/><a NAME="E4:9_1_2_1"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">ij</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">ij</font> <a href="mcart_1.html#K1">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">ij</font> <a href="mcart_1.html#K2">`2</a> </span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, <a class="ref" href="mcart_1.html#T23">MCART_1:23</a></i>;<br/><a NAME="E5:9_1_2_1"/><b>then </b><i><font color="Green">E101</font></i>: 
<font color="Maroon">i2</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">i1</font>
 <b>by </b><i>, <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/><a NAME="E6:9_1_2_1"/>
<font color="Maroon">N1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">ij</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N1</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">i2</font>,<font color="Maroon">i1</font>
 <b>by </b><i/>;<br/><a NAME="E7:9_1_2_1"/><b>then </b><i><font color="Green">E102</font></i>: 
<font color="Maroon">N1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">ij</font> <a href="hidden.html#R1">=</a>  <a href="cqc_lang.html#K1">IFEQ</a> <font color="Maroon">i2</font>,<font color="Maroon">i1</font>,<span class="p1">(<span class="default"><a href="msualg_3.html#K2">id</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p2">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i1</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msalimit.html#K1" target="_self">bind</a> <font color="Maroon">B</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="msualg_3.html#K3">**</a> <span class="p2">(<span class="default"><a href="msualg_3.html#K2">id</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p3">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i1</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E8:9_1_2_1"/>
<font color="Maroon">N2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">ij</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">i2</font>,<font color="Maroon">i1</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E9:9_1_2_1"/>
<font color="Maroon">N1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">ij</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">ij</font>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:9_1_2"/>
<font color="Maroon">N1</font> <a href="pboole.html#R6">=</a> <font color="Maroon">N2</font>
 <b>by </b><i><a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>;<br/>


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