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<span class="kw">let </span><font color="Maroon" title="c5">N1</font>, <font color="Maroon" title="c6">N2</font> be    <a href="msalimit.html#M2" title="MSALIMIT:mode.2">Binding</a> of <font color="Maroon" title="c3">OAF</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">P</font>  st <font color="Olive" title="b1">i</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Olive" title="b2">j</font> holds <br/><font color="Maroon" title="c5">N1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b2">j</font>,<font color="Olive" title="b1">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funcop_1.html#K15" title="FUNCOP_1:func.15">IFEQ</a> (<font color="Olive" title="b2">j</font>,<font color="Olive" title="b1">i</font>,<span class="p1">(<span class="default"><a href="msualg_3.html#K2" title="MSUALG_3:func.2">id</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c3">OAF</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Olive" title="b1">i</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msalimit.html#K1" title="MSALIMIT:func.1">bind</a> (<font color="Maroon" title="c4">B</font>,<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>)</span>)</span> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <span class="p2">(<span class="default"><a href="msualg_3.html#K2" title="MSUALG_3:func.2">id</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p3">(<span class="default"><font color="Maroon" title="c3">OAF</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Olive" title="b1">i</font></span>)</span></span>)</span></span>)</span>) ) &amp; (  for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">P</font>  st <font color="Olive" title="b1">i</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Olive" title="b2">j</font> holds <br/><font color="Maroon" title="c6">N2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b2">j</font>,<font color="Olive" title="b1">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funcop_1.html#K15" title="FUNCOP_1:func.15">IFEQ</a> (<font color="Olive" title="b2">j</font>,<font color="Olive" title="b1">i</font>,<span class="p1">(<span class="default"><a href="msualg_3.html#K2" title="MSUALG_3:func.2">id</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c3">OAF</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Olive" title="b1">i</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msalimit.html#K1" title="MSALIMIT:func.1">bind</a> (<font color="Maroon" title="c4">B</font>,<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>)</span>)</span> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <span class="p2">(<span class="default"><a href="msualg_3.html#K2" title="MSUALG_3:func.2">id</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p3">(<span class="default"><font color="Maroon" title="c3">OAF</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Olive" title="b1">i</font></span>)</span></span>)</span></span>)</span>) ) implies <font color="Maroon" title="c5">N1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">N2</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:9_1_2"/><span class="lab"><font color="Green" title="E7">A39</font></span>: 
 for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">P</font>  st <font color="Olive" title="b1">i</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Olive" title="b2">j</font> holds <br/><font color="Maroon" title="c5">N1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b2">j</font>,<font color="Olive" title="b1">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funcop_1.html#K15" title="FUNCOP_1:func.15">IFEQ</a> (<font color="Olive" title="b2">j</font>,<font color="Olive" title="b1">i</font>,<span class="p1">(<span class="default"><a href="msualg_3.html#K2" title="MSUALG_3:func.2">id</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c3">OAF</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Olive" title="b1">i</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msalimit.html#K1" title="MSALIMIT:func.1">bind</a> (<font color="Maroon" title="c4">B</font>,<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>)</span>)</span> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <span class="p2">(<span class="default"><a href="msualg_3.html#K2" title="MSUALG_3:func.2">id</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p3">(<span class="default"><font color="Maroon" title="c3">OAF</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Olive" title="b1">i</font></span>)</span></span>)</span></span>)</span>)
 <span class="kw">and </span><br/><a NAME="E2:9_1_2"/><span class="lab"><font color="Green" title="E8">A40</font></span>: 
 for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">P</font>  st <font color="Olive" title="b1">i</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Olive" title="b2">j</font> holds <br/><font color="Maroon" title="c6">N2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b2">j</font>,<font color="Olive" title="b1">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funcop_1.html#K15" title="FUNCOP_1:func.15">IFEQ</a> (<font color="Olive" title="b2">j</font>,<font color="Olive" title="b1">i</font>,<span class="p1">(<span class="default"><a href="msualg_3.html#K2" title="MSUALG_3:func.2">id</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c3">OAF</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Olive" title="b1">i</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msalimit.html#K1" title="MSALIMIT:func.1">bind</a> (<font color="Maroon" title="c4">B</font>,<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>)</span>)</span> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <span class="p2">(<span class="default"><a href="msualg_3.html#K2" title="MSUALG_3:func.2">id</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p3">(<span class="default"><font color="Maroon" title="c3">OAF</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Olive" title="b1">i</font></span>)</span></span>)</span></span>)</span>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">N1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">N2</font></span><br/>

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="9_1_2_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">ij</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">ij</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">P</font> holds <br/><font color="Maroon" title="c5">N1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">ij</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">N2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">ij</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">ij</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">ij</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">P</font> implies <font color="Maroon" title="c5">N1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">ij</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">N2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">ij</font> )</span><br/><span class="kw">assume </span><a NAME="E1:9_1_2_1"/><span class="lab"><font color="Green" title="E9">A41</font></span>: 
<font color="Maroon" title="c7">ij</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">P</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">N1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">ij</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">N2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">ij</font></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">i2</font> = <font color="Maroon" title="c7">ij</font> <a href="xtuple_0.html#K1" title="XTUPLE_0:func.1">`1</a> , <font color="Maroon" title="c9">i1</font> = <font color="Maroon" title="c7">ij</font> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a>  as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">P</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T10" onmouseover="rs('mcart_1/T10')" onmouseout="rh()">MCART_1:10</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c10">i1</font> = <font color="Maroon" title="c9">i1</font>, <font color="Maroon" title="c11">i2</font> = <font color="Maroon" title="c8">i2</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">P</font> ;<br/><a NAME="E4:9_1_2_1"/><span class="lab"><font color="Green" title="E10">A42</font></span>: 
<font color="Maroon" title="c6">N2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">ij</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">N2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c11">i2</font>,<font color="Maroon" title="c10">i1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9_1_2_1"><span class="lab"><font color="Green" title="E9">A41</font></span></a>, <a class="ref" href="mcart_1.html#T21" onmouseover="rs('mcart_1/T21')" onmouseout="rh()">MCART_1:21</a></span>;<br/><a NAME="E5:9_1_2_1"/>
<font color="Maroon" title="c7">ij</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">ij</font> <a href="xtuple_0.html#K1" title="XTUPLE_0:func.1">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">ij</font> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9_1_2_1"><span class="lab"><font color="Green" title="E9">A41</font></span></a>, <a class="ref" href="mcart_1.html#T21" onmouseover="rs('mcart_1/T21')" onmouseout="rh()">MCART_1:21</a></span>;<br/><a NAME="E6:9_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E11">A43</font></span>: 
<font color="Maroon" title="c11">i2</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c10">i1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9_1_2_1"><span class="lab"><font color="Green" title="E9">A41</font></span></a>, <a class="ref" href="orders_2.html#D5" onmouseover="rs('orders_2/D5')" onmouseout="rh()">ORDERS_2:def 5</a></span>;<br/><a NAME="E7:9_1_2_1"/>
<font color="Maroon" title="c5">N1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">ij</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">N1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c11">i2</font>,<font color="Maroon" title="c10">i1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9_1_2_1"><span class="lab"><font color="Green" title="E9">A41</font></span></a>, <a class="ref" href="mcart_1.html#T21" onmouseover="rs('mcart_1/T21')" onmouseout="rh()">MCART_1:21</a></span>;<br/><a NAME="E8:9_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">N1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">ij</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funcop_1.html#K15" title="FUNCOP_1:func.15">IFEQ</a> (<font color="Maroon" title="c11">i2</font>,<font color="Maroon" title="c10">i1</font>,<span class="p1">(<span class="default"><a href="msualg_3.html#K2" title="MSUALG_3:func.2">id</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c3">OAF</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c10">i1</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msalimit.html#K1" title="MSALIMIT:func.1">bind</a> (<font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c10">i1</font>,<font color="Maroon" title="c11">i2</font>)</span>)</span> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <span class="p2">(<span class="default"><a href="msualg_3.html#K2" title="MSUALG_3:func.2">id</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p3">(<span class="default"><font color="Maroon" title="c3">OAF</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c10">i1</font></span>)</span></span>)</span></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9_1_2"><span class="lab"><font color="Green" title="E7">A39</font></span></a>, <a class="txt" href="#E6:9_1_2_1"><span class="lab"><font color="Green" title="E11">A43</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E9:9_1_2_1"/>
<font color="Maroon" title="c5">N1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">ij</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">N2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">ij</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:9_1_2"><span class="lab"><font color="Green" title="E8">A40</font></span></a>, <a class="txt" href="#E6:9_1_2_1"><span class="lab"><font color="Green" title="E11">A43</font></span></a>, <a class="txt" href="#E4:9_1_2_1"><span class="lab"><font color="Green" title="E10">A42</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:9_1_2"/>
<font color="Maroon" title="c5">N1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c6">N2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#T3" onmouseover="rs('pboole/T3')" onmouseout="rh()">PBOOLE:3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
