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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <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 <br/>( $1 <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"><font color="Olive" title="b2">j</font>,<font color="Olive" title="b1">i</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">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> );<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">ij</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">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  ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ex <font color="Olive" title="b2">i1</font>, <font color="Olive" title="b3">i2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">P</font> st <br/>( <font color="Maroon" title="c5">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"><font color="Olive" title="b3">i2</font>,<font color="Olive" title="b2">i1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">IFEQ</a> <font color="Olive" title="b3">i2</font>,<font color="Olive" title="b2">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="Olive" title="b2">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="Olive" title="b2">i1</font>,<font color="Olive" title="b3">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="Olive" title="b2">i1</font></span>)</span></span>)</span></span>)</span> ) )</span><br/><b>assume </b><a NAME="E1:9_1_1_1"/><i><font color="Green" title="E7">A1</font></i>: 
<font color="Maroon" title="c5">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" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ex <font color="Olive" title="b2">i1</font>, <font color="Olive" title="b3">i2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">P</font> st <br/>( <font color="Maroon" title="c5">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"><font color="Olive" title="b3">i2</font>,<font color="Olive" title="b2">i1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">IFEQ</a> <font color="Olive" title="b3">i2</font>,<font color="Olive" title="b2">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="Olive" title="b2">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="Olive" title="b2">i1</font>,<font color="Olive" title="b3">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="Olive" title="b2">i1</font></span>)</span></span>)</span></span>)</span> )</span><br/><b>then reconsider </b><font color="Maroon" title="c6">i2</font> = <font color="Maroon" title="c5">ij</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> , <font color="Maroon" title="c7">i1</font> = <font color="Maroon" title="c5">ij</font> <a href="mcart_1.html#K2" title="MCART_1: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> <b>by </b><i><a class="ref" href="mcart_1.html#T10" title="MCART_1:th.10">MCART_1:10</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c8">i1</font> = <font color="Maroon" title="c7">i1</font>, <font color="Maroon" title="c9">i2</font> = <font color="Maroon" title="c6">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/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  =  <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">IFEQ</a> <font color="Maroon" title="c9">i2</font>,<font color="Maroon" title="c8">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="c8">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="c8">i1</font>,<font color="Maroon" title="c9">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="c8">i1</font></span>)</span></span>)</span></span>)</span>;<br/><b>consider </b><font color="Maroon" title="c10">A</font> being    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">P</font><b> such that </b><br/><a NAME="E5:9_1_1_1"/><i><font color="Green" title="E8">A2</font></i>: 
for <font color="Olive" title="b1">ij</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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="c10">A</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">H<sub>1</sub></font>(<font color="Olive" title="b1">ij</font>)
 <b>from </b><i><a class="ref" href="pboole.html#S4" title="PBOOLE:sch.4">PBOOLE:sch 4</a>();<br/></i><b>take </b><font color="Maroon" title="c11">x</font> = <font color="Maroon" title="c10">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">ij</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">i1</font>, <font color="Olive" title="b2">i2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">P</font> st <br/>( <font color="Maroon" title="c5">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"><font color="Olive" title="b2">i2</font>,<font color="Olive" title="b1">i1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">IFEQ</a> <font color="Olive" title="b2">i2</font>,<font color="Olive" title="b1">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="Olive" title="b1">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="Olive" title="b1">i1</font>,<font color="Olive" title="b2">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="Olive" title="b1">i1</font></span>)</span></span>)</span></span>)</span> )</span><br/><b>take </b><font color="Maroon" title="c12">i1</font> = <font color="Maroon" title="c8">i1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">i2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">P</font> st <br/>( <font color="Maroon" title="c5">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"><font color="Olive" title="b1">i2</font>,<font color="Maroon" title="c12">i1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">IFEQ</a> <font color="Olive" title="b1">i2</font>,<font color="Maroon" title="c12">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="c12">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="c12">i1</font>,<font color="Olive" title="b1">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="c12">i1</font></span>)</span></span>)</span></span>)</span> )</span><br/><b>take </b><font color="Maroon" title="c13">i2</font> = <font color="Maroon" title="c9">i2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">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"><font color="Maroon" title="c13">i2</font>,<font color="Maroon" title="c12">i1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">IFEQ</a> <font color="Maroon" title="c13">i2</font>,<font color="Maroon" title="c12">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="c12">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="c12">i1</font>,<font color="Maroon" title="c13">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="c12">i1</font></span>)</span></span>)</span></span>)</span> )</span><br/><b>thus </b><a NAME="E6:9_1_1_1"/>
( <font color="Maroon" title="c5">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"><font color="Maroon" title="c13">i2</font>,<font color="Maroon" title="c12">i1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">IFEQ</a> <font color="Maroon" title="c13">i2</font>,<font color="Maroon" title="c12">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="c12">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="c12">i1</font>,<font color="Maroon" title="c13">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="c12">i1</font></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="#E1:9_1_1_1"><i><font color="Green" title="E7">A1</font></i></a>, <a class="txt" href="#E5:9_1_1_1"><i><font color="Green" title="E8">A2</font></i></a>, <a class="ref" href="mcart_1.html#T23" title="MCART_1:th.23">MCART_1:23</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E2:9_1_1"/><b>then </b><i><font color="Green" title="E7">A3</font></i>: 
for <font color="Olive" title="b1">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">z</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/> ex <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">z</font>,<font color="Olive" title="b2">y</font>]
 
;<br/>
<b>consider </b><font color="Maroon" title="c5">A</font> being    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">P</font><b> such that </b><br/><a NAME="E4:9_1_1"/><i><font color="Green" title="E8">A4</font></i>: 
for <font color="Olive" title="b1">ij</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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">S<sub>1</sub></font>[<font color="Olive" title="b1">ij</font>,<font color="Maroon" title="c5">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">ij</font>]
 <b>from </b><i><a class="ref" href="pboole.html#S3" title="PBOOLE:sch.3">PBOOLE:sch 3</a>(<a class="txt" href="#E2:9_1_1"><i><font color="Green" title="E7">A3</font></i></a>);<br/></i>
<a NAME="E5:9_1_1"/>
for <font color="Olive" title="b1">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">A</font> holds <br/><font color="Maroon" title="c5">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font> is   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">A</font> implies <font color="Maroon" title="c5">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">z</font> is   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> )</span><br/>

<b>assume </b><a NAME="E1:9_1_1_2"/>
<font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">z</font> is   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a></span><br/>

<a NAME="E2:9_1_1_2"/><b>then </b>
<font color="Maroon" title="c6">z</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>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c7">i1</font>, <font color="Maroon" title="c8">i2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">P</font><b> such that </b><br/><a NAME="E4:9_1_1_2"/><i><font color="Green" title="E9">A5</font></i>: 
( <font color="Maroon" title="c6">z</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"><font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c7">i1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c5">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">IFEQ</a> <font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c7">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="c7">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="c7">i1</font>,<font color="Maroon" title="c8">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="c7">i1</font></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="#E4:9_1_1"><i><font color="Green" title="E8">A4</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:9_1_1_2_1"/>
( <font color="Maroon" title="c7">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">i2</font> or <font color="Maroon" title="c7">i1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">i2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:9_1_1_2_1_1"/>
<font color="Maroon" title="c7">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">i2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">z</font> is   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a></span><br/><div class="add"><b>hence </b><a NAME="E2:9_1_1_2_1_1"/>
<font color="Maroon" title="c5">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">z</font> is   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>
 <b>by </b><i><a class="txt" href="#E4:9_1_1_2"><i><font color="Green" title="E9">A5</font></i></a>, <a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:9_1_1_2_1_2"/>
<font color="Maroon" title="c7">i1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">i2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">z</font> is   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a></span><br/><div class="add"><b>hence </b><a NAME="E2:9_1_1_2_1_2"/>
<font color="Maroon" title="c5">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">z</font> is   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>
 <b>by </b><i><a class="txt" href="#E4:9_1_1_2"><i><font color="Green" title="E9">A5</font></i></a>, <a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c6">A</font> = <font color="Maroon" title="c5">A</font> as   <a href="pboole.html#NM2" title="PBOOLE:NM.2">ManySortedFunction</a> of the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">P</font> <b>by </b><i><a class="ref" href="funcop_1.html#D6" title="FUNCOP_1:def.6">FUNCOP_1:def 6</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">i</font>, <font color="Maroon" title="c8">j</font>, <font color="Maroon" title="c9">k</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">P</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">i</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c8">j</font> &amp; <font color="Maroon" title="c8">j</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c9">k</font> implies  ex <font color="Olive" title="b1">f1</font> being  <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <span class="p1">(<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="c7">i</font></span>)</span>,<span class="p1">(<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="c8">j</font></span>)</span> ex <font color="Olive" title="b2">f2</font> being  <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <span class="p1">(<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="c8">j</font></span>)</span>,<span class="p1">(<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="c9">k</font></span>)</span> st <br/>( <font color="Olive" title="b1">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">A</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c7">i</font> &amp; <font color="Olive" title="b2">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">A</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">k</font>,<font color="Maroon" title="c8">j</font> &amp; <font color="Maroon" title="c6">A</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">k</font>,<font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f2</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Olive" title="b1">f1</font> &amp; <font color="Olive" title="b1">f1</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c3">OAF</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c3">OAF</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c8">j</font> ) )</span><br/><b>assume </b><a NAME="E1:9_1_1_3"/><i><font color="Green" title="E9">A6</font></i>: 
( <font color="Maroon" title="c7">i</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c8">j</font> &amp; <font color="Maroon" title="c8">j</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c9">k</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">f1</font> being  <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <span class="p1">(<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="c7">i</font></span>)</span>,<span class="p1">(<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="c8">j</font></span>)</span> ex <font color="Olive" title="b2">f2</font> being  <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <span class="p1">(<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="c8">j</font></span>)</span>,<span class="p1">(<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="c9">k</font></span>)</span> st <br/>( <font color="Olive" title="b1">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">A</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c7">i</font> &amp; <font color="Olive" title="b2">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">A</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">k</font>,<font color="Maroon" title="c8">j</font> &amp; <font color="Maroon" title="c6">A</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">k</font>,<font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f2</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Olive" title="b1">f1</font> &amp; <font color="Olive" title="b1">f1</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c3">OAF</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c3">OAF</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c8">j</font> )</span><br/><b>consider </b><font color="Maroon" title="c10">kl</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:9_1_1_3"/><i><font color="Green" title="E10">A7</font></i>: 
<font color="Maroon" title="c10">kl</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"><font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c7">i</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 ;<br/><a NAME="E4:9_1_1_3"/>
<font color="Maroon" title="c10">kl</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>
 <b>by </b><i><a class="txt" href="#E1:9_1_1_3"><i><font color="Green" title="E9">A6</font></i></a>, <a class="txt" href="#E3:9_1_1_3"><i><font color="Green" title="E10">A7</font></i></a>, <a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/><b>then consider </b><font color="Maroon" title="c11">i1</font>, <font color="Maroon" title="c12">j1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">P</font><b> such that </b><br/><a NAME="E6:9_1_1_3"/><i><font color="Green" title="E11">A8</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">j1</font>,<font color="Maroon" title="c11">i1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">kl</font> &amp; <font color="Maroon" title="c6">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">kl</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">IFEQ</a> <font color="Maroon" title="c12">j1</font>,<font color="Maroon" title="c11">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="c11">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="c11">i1</font>,<font color="Maroon" title="c12">j1</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="c11">i1</font></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="#E4:9_1_1"><i><font color="Green" title="E8">A4</font></i></a></i>;<br/><a NAME="E7:9_1_1_3"/><i><font color="Green" title="E12">A9</font></i>: 
( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">i</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">j</font> )
 <b>by </b><i><a class="txt" href="#E3:9_1_1_3"><i><font color="Green" title="E10">A7</font></i></a>, <a class="txt" href="#E6:9_1_1_3"><i><font color="Green" title="E11">A8</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><a NAME="E8:9_1_1_3"/>
<font color="Maroon" title="c6">A</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c7">i</font> is   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <span class="p1">(<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="c7">i</font></span>)</span>,<span class="p1">(<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="c8">j</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_3_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:9_1_1_3_1_1"/>
( <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">j</font> or <font color="Maroon" title="c7">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">j</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:9_1_1_3_1_1_1"/>
<font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">A</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c7">i</font> is   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <span class="p1">(<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="c7">i</font></span>)</span>,<span class="p1">(<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="c8">j</font></span>)</span></span><br/><div class="add"><b>hence </b><a NAME="E2:9_1_1_3_1_1_1"/>
<font color="Maroon" title="c6">A</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c7">i</font> is   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <span class="p1">(<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="c7">i</font></span>)</span>,<span class="p1">(<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="c8">j</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E6:9_1_1_3"><i><font color="Green" title="E11">A8</font></i></a>, <a class="txt" href="#E7:9_1_1_3"><i><font color="Green" title="E12">A9</font></i></a>, <a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:9_1_1_3_1_1_2"/>
<font color="Maroon" title="c7">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">A</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c7">i</font> is   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <span class="p1">(<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="c7">i</font></span>)</span>,<span class="p1">(<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="c8">j</font></span>)</span></span><br/><div class="add"><b>hence </b><a NAME="E2:9_1_1_3_1_1_2"/>
<font color="Maroon" title="c6">A</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c7">i</font> is   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <span class="p1">(<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="c7">i</font></span>)</span>,<span class="p1">(<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="c8">j</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E6:9_1_1_3"><i><font color="Green" title="E11">A8</font></i></a>, <a class="txt" href="#E7:9_1_1_3"><i><font color="Green" title="E12">A9</font></i></a>, <a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c13">f1</font> = <font color="Maroon" title="c6">A</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c7">i</font> as   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <span class="p1">(<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="c7">i</font></span>)</span>,<span class="p1">(<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="c8">j</font></span>)</span> ;<br/><b>consider </b><font color="Maroon" title="c14">lm</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E11:9_1_1_3"/><i><font color="Green" title="E13">A10</font></i>: 
<font color="Maroon" title="c14">lm</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"><font color="Maroon" title="c9">k</font>,<font color="Maroon" title="c8">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 ;<br/><a NAME="E12:9_1_1_3"/>
<font color="Maroon" title="c14">lm</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>
 <b>by </b><i><a class="txt" href="#E1:9_1_1_3"><i><font color="Green" title="E9">A6</font></i></a>, <a class="txt" href="#E11:9_1_1_3"><i><font color="Green" title="E13">A10</font></i></a>, <a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/><b>then consider </b><font color="Maroon" title="c15">i2</font>, <font color="Maroon" title="c16">j2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">P</font><b> such that </b><br/><a NAME="E14:9_1_1_3"/><i><font color="Green" title="E14">A11</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">j2</font>,<font color="Maroon" title="c15">i2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">lm</font> &amp; <font color="Maroon" title="c6">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">lm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">IFEQ</a> <font color="Maroon" title="c16">j2</font>,<font color="Maroon" title="c15">i2</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="c15">i2</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="c15">i2</font>,<font color="Maroon" title="c16">j2</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="c15">i2</font></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="#E4:9_1_1"><i><font color="Green" title="E8">A4</font></i></a></i>;<br/><a NAME="E15:9_1_1_3"/><i><font color="Green" title="E15">A12</font></i>: 
( <font color="Maroon" title="c16">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">k</font> &amp; <font color="Maroon" title="c15">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">j</font> )
 <b>by </b><i><a class="txt" href="#E11:9_1_1_3"><i><font color="Green" title="E13">A10</font></i></a>, <a class="txt" href="#E14:9_1_1_3"><i><font color="Green" title="E14">A11</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><a NAME="E16:9_1_1_3"/>
<font color="Maroon" title="c6">A</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">k</font>,<font color="Maroon" title="c8">j</font> is   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <span class="p1">(<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="c8">j</font></span>)</span>,<span class="p1">(<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="c9">k</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_3_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_3_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:9_1_1_3_2_1"/>
( <font color="Maroon" title="c8">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">k</font> or <font color="Maroon" title="c8">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">k</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_3_2_1_1"><b>suppose </b></a><a NAME="E1:9_1_1_3_2_1_1"/>
<font color="Maroon" title="c8">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">A</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">k</font>,<font color="Maroon" title="c8">j</font> is   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <span class="p1">(<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="c8">j</font></span>)</span>,<span class="p1">(<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="c9">k</font></span>)</span></span><br/><div class="add"><b>hence </b><a NAME="E2:9_1_1_3_2_1_1"/>
<font color="Maroon" title="c6">A</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">k</font>,<font color="Maroon" title="c8">j</font> is   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <span class="p1">(<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="c8">j</font></span>)</span>,<span class="p1">(<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="c9">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E14:9_1_1_3"><i><font color="Green" title="E14">A11</font></i></a>, <a class="txt" href="#E15:9_1_1_3"><i><font color="Green" title="E15">A12</font></i></a>, <a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_3_2_1_2"><b>suppose </b></a><a NAME="E1:9_1_1_3_2_1_2"/>
<font color="Maroon" title="c8">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">A</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">k</font>,<font color="Maroon" title="c8">j</font> is   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <span class="p1">(<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="c8">j</font></span>)</span>,<span class="p1">(<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="c9">k</font></span>)</span></span><br/><div class="add"><b>hence </b><a NAME="E2:9_1_1_3_2_1_2"/>
<font color="Maroon" title="c6">A</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">k</font>,<font color="Maroon" title="c8">j</font> is   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <span class="p1">(<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="c8">j</font></span>)</span>,<span class="p1">(<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="c9">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E14:9_1_1_3"><i><font color="Green" title="E14">A11</font></i></a>, <a class="txt" href="#E15:9_1_1_3"><i><font color="Green" title="E15">A12</font></i></a>, <a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c17">f2</font> = <font color="Maroon" title="c6">A</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">k</font>,<font color="Maroon" title="c8">j</font> as   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <span class="p1">(<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="c8">j</font></span>)</span>,<span class="p1">(<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="c9">k</font></span>)</span> ;<br/><a NAME="E18:9_1_1_3"/><i><font color="Green" title="E16">A13</font></i>: 
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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">j</font> holds <br/><font color="Maroon" title="c6">A</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="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="p1">(<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>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_3_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c18">i</font>, <font color="Maroon" title="c19">j</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">P</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c18">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j</font> implies <font color="Maroon" title="c6">A</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c19">j</font>,<font color="Maroon" title="c18">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="p1">(<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="c18">i</font></span>)</span> )</span><br/>



<b>assume </b><a NAME="E1:9_1_1_3_3"/><i><font color="Green" title="E17">A14</font></i>: 
<font color="Maroon" title="c18">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">A</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c19">j</font>,<font color="Maroon" title="c18">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="p1">(<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="c18">i</font></span>)</span></span><br/>

<a NAME="E2:9_1_1_3_3"/><b>then </b><i><font color="Green" title="E18">A15</font></i>: 
<font color="Maroon" title="c18">i</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c19">j</font>
 
<b>by </b><i><a class="ref" href="orders_2.html#T24" title="ORDERS_2:th.24">ORDERS_2:24</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c20">lm</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:9_1_1_3_3"/><i><font color="Green" title="E19">A16</font></i>: 
<font color="Maroon" title="c20">lm</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"><font color="Maroon" title="c19">j</font>,<font color="Maroon" title="c18">i</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 ;<br/>
<a NAME="E5:9_1_1_3_3"/>
<font color="Maroon" title="c20">lm</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>
 
<b>by </b><i><a class="txt" href="#E2:9_1_1_3_3"><i><font color="Green" title="E18">A15</font></i></a>, <a class="txt" href="#E4:9_1_1_3_3"><i><font color="Green" title="E19">A16</font></i></a>, <a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c21">i2</font>, <font color="Maroon" title="c22">j2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">P</font><b> such that </b><br/><a NAME="E7:9_1_1_3_3"/><i><font color="Green" title="E20">A17</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c22">j2</font>,<font color="Maroon" title="c21">i2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">lm</font> &amp; <font color="Maroon" title="c6">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">lm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">IFEQ</a> <font color="Maroon" title="c22">j2</font>,<font color="Maroon" title="c21">i2</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="c21">i2</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="c21">i2</font>,<font color="Maroon" title="c22">j2</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="c21">i2</font></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="#E4:9_1_1"><i><font color="Green" title="E8">A4</font></i></a></i>;<br/>
<a NAME="E8:9_1_1_3_3"/>
( <font color="Maroon" title="c21">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i</font> &amp; <font color="Maroon" title="c22">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j</font> )
 
<b>by </b><i><a class="txt" href="#E4:9_1_1_3_3"><i><font color="Green" title="E19">A16</font></i></a>, <a class="txt" href="#E7:9_1_1_3_3"><i><font color="Green" title="E20">A17</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E9:9_1_1_3_3"/>
<font color="Maroon" title="c6">A</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c19">j</font>,<font color="Maroon" title="c18">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="p1">(<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="c18">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:9_1_1_3_3"><i><font color="Green" title="E17">A14</font></i></a>, <a class="txt" href="#E7:9_1_1_3_3"><i><font color="Green" title="E20">A17</font></i></a>, <a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E19:9_1_1_3"/><i><font color="Green" title="E17">A18</font></i>: 
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> &amp; <font color="Olive" title="b1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">j</font> holds <br/><font color="Maroon" title="c6">A</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="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>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_3_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c18">i</font>, <font color="Maroon" title="c19">j</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">P</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c18">i</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c19">j</font> &amp; <font color="Maroon" title="c18">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c19">j</font> implies <font color="Maroon" title="c6">A</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c19">j</font>,<font color="Maroon" title="c18">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msalimit.html#K1" title="MSALIMIT:func.1">bind</a> <font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c18">i</font>,<font color="Maroon" title="c19">j</font> )</span><br/>



<b>assume </b><a NAME="E1:9_1_1_3_4"/><i><font color="Green" title="E18">A19</font></i>: 
( <font color="Maroon" title="c18">i</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c19">j</font> &amp; <font color="Maroon" title="c18">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c19">j</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">A</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c19">j</font>,<font color="Maroon" title="c18">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msalimit.html#K1" title="MSALIMIT:func.1">bind</a> <font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c18">i</font>,<font color="Maroon" title="c19">j</font></span><br/>

<b>consider </b><font color="Maroon" title="c20">lm</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:9_1_1_3_4"/><i><font color="Green" title="E19">A20</font></i>: 
<font color="Maroon" title="c20">lm</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"><font color="Maroon" title="c19">j</font>,<font color="Maroon" title="c18">i</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 ;<br/>
<a NAME="E4:9_1_1_3_4"/>
<font color="Maroon" title="c20">lm</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>
 
<b>by </b><i><a class="txt" href="#E1:9_1_1_3_4"><i><font color="Green" title="E18">A19</font></i></a>, <a class="txt" href="#E3:9_1_1_3_4"><i><font color="Green" title="E19">A20</font></i></a>, <a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c21">i2</font>, <font color="Maroon" title="c22">j2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">P</font><b> such that </b><br/><a NAME="E6:9_1_1_3_4"/><i><font color="Green" title="E20">A21</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c22">j2</font>,<font color="Maroon" title="c21">i2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">lm</font> &amp; <font color="Maroon" title="c6">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">lm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">IFEQ</a> <font color="Maroon" title="c22">j2</font>,<font color="Maroon" title="c21">i2</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="c21">i2</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="c21">i2</font>,<font color="Maroon" title="c22">j2</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="c21">i2</font></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="#E4:9_1_1"><i><font color="Green" title="E8">A4</font></i></a></i>;<br/>
<a NAME="E7:9_1_1_3_4"/>
( <font color="Maroon" title="c21">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i</font> &amp; <font color="Maroon" title="c22">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j</font> )
 
<b>by </b><i><a class="txt" href="#E3:9_1_1_3_4"><i><font color="Green" title="E19">A20</font></i></a>, <a class="txt" href="#E6:9_1_1_3_4"><i><font color="Green" title="E20">A21</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E8:9_1_1_3_4"/><b>then </b>
<font color="Maroon" title="c6">A</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c19">j</font>,<font color="Maroon" title="c18">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<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="c18">i</font>,<font color="Maroon" title="c19">j</font></span>)</span> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <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="c18">i</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:9_1_1_3_4"><i><font color="Green" title="E18">A19</font></i></a>, <a class="txt" href="#E6:9_1_1_3_4"><i><font color="Green" title="E20">A21</font></i></a>, <a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></i>;<br/>
<b>hence </b><a NAME="E9:9_1_1_3_4"/>
<font color="Maroon" title="c6">A</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c19">j</font>,<font color="Maroon" title="c18">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msalimit.html#K1" title="MSALIMIT:func.1">bind</a> <font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c18">i</font>,<font color="Maroon" title="c19">j</font>
 <b>by </b><i><a class="ref" href="msualg_3.html#T3" title="MSUALG_3:th.3">MSUALG_3:3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E20:9_1_1_3"/><i><font color="Green" title="E18">A22</font></i>: 
<font color="Maroon" title="c6">A</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">k</font>,<font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">f2</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c13">f1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_3_5"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_3_5_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:9_1_1_3_5_1"/>
( ( <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">j</font> &amp; <font color="Maroon" title="c8">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">k</font> ) or ( <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">j</font> &amp; <font color="Maroon" title="c8">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">k</font> ) or ( <font color="Maroon" title="c7">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">j</font> &amp; <font color="Maroon" title="c8">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">k</font> ) or ( <font color="Maroon" title="c7">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">j</font> &amp; <font color="Maroon" title="c8">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">k</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_3_5_1_1"><b>suppose </b></a><a NAME="E1:9_1_1_3_5_1_1"/><i><font color="Green" title="E19">A23</font></i>: 
( <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">j</font> &amp; <font color="Maroon" title="c8">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">k</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">A</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">k</font>,<font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">f2</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c13">f1</font></span><br/><div class="add"><a NAME="E2:9_1_1_3_5_1_1"/><b>then </b>
<font color="Maroon" title="c17">f2</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <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="p1">(<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="c8">j</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E14:9_1_1_3"><i><font color="Green" title="E14">A11</font></i></a>, <a class="txt" href="#E15:9_1_1_3"><i><font color="Green" title="E15">A12</font></i></a>, <a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></i>;<br/><b>hence </b><a NAME="E3:9_1_1_3_5_1_1"/>
<font color="Maroon" title="c6">A</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">k</font>,<font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">f2</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c13">f1</font>
 <b>by </b><i><a class="txt" href="#E1:9_1_1_3_5_1_1"><i><font color="Green" title="E19">A23</font></i></a>, <a class="ref" href="msualg_3.html#T3" title="MSUALG_3:th.3">MSUALG_3:3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_3_5_1_2"><b>suppose </b></a><a NAME="E1:9_1_1_3_5_1_2"/><i><font color="Green" title="E19">A24</font></i>: 
( <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">j</font> &amp; <font color="Maroon" title="c8">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">k</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">A</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">k</font>,<font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">f2</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c13">f1</font></span><br/><div class="add"><a NAME="E2:9_1_1_3_5_1_2"/><b>then </b>
<font color="Maroon" title="c13">f1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <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="p1">(<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="c7">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E6:9_1_1_3"><i><font color="Green" title="E11">A8</font></i></a>, <a class="txt" href="#E7:9_1_1_3"><i><font color="Green" title="E12">A9</font></i></a>, <a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></i>;<br/><b>hence </b><a NAME="E3:9_1_1_3_5_1_2"/>
<font color="Maroon" title="c6">A</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">k</font>,<font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">f2</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c13">f1</font>
 <b>by </b><i><a class="txt" href="#E1:9_1_1_3_5_1_2"><i><font color="Green" title="E19">A24</font></i></a>, <a class="ref" href="msualg_3.html#T3" title="MSUALG_3:th.3">MSUALG_3:3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_3_5_1_3"><b>suppose </b></a><a NAME="E1:9_1_1_3_5_1_3"/><i><font color="Green" title="E19">A25</font></i>: 
( <font color="Maroon" title="c7">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">j</font> &amp; <font color="Maroon" title="c8">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">k</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">A</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">k</font>,<font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">f2</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c13">f1</font></span><br/><div class="add"><a NAME="E2:9_1_1_3_5_1_3"/><b>then </b>
<font color="Maroon" title="c17">f2</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <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="p1">(<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="c8">j</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E14:9_1_1_3"><i><font color="Green" title="E14">A11</font></i></a>, <a class="txt" href="#E15:9_1_1_3"><i><font color="Green" title="E15">A12</font></i></a>, <a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></i>;<br/><b>hence </b><a NAME="E3:9_1_1_3_5_1_3"/>
<font color="Maroon" title="c6">A</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">k</font>,<font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">f2</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c13">f1</font>
 <b>by </b><i><a class="txt" href="#E1:9_1_1_3_5_1_3"><i><font color="Green" title="E19">A25</font></i></a>, <a class="ref" href="msualg_3.html#T4" title="MSUALG_3:th.4">MSUALG_3:4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_3_5_1_4"><b>suppose </b></a><a NAME="E1:9_1_1_3_5_1_4"/><i><font color="Green" title="E19">A26</font></i>: 
( <font color="Maroon" title="c7">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">j</font> &amp; <font color="Maroon" title="c8">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">k</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">A</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">k</font>,<font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">f2</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c13">f1</font></span><br/><div class="add"><a NAME="E2:9_1_1_3_5_1_4"/><b>then </b>
( <font color="Maroon" title="c7">i</font> <a href="orders_2.html#NR4" title="ORDERS_2:NR.4">&gt;</a> <font color="Maroon" title="c8">j</font> &amp; <font color="Maroon" title="c8">j</font> <a href="orders_2.html#NR4" title="ORDERS_2:NR.4">&gt;</a> <font color="Maroon" title="c9">k</font> )
 <b>by </b><i><a class="txt" href="#E1:9_1_1_3"><i><font color="Green" title="E9">A6</font></i></a>, <a class="ref" href="orders_2.html#D10" title="ORDERS_2:def.10">ORDERS_2:def 10</a></i>;<br/><a NAME="E3:9_1_1_3_5_1_4"/><b>then </b><i><font color="Green" title="E20">A27</font></i>: 
<font color="Maroon" title="c7">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">k</font>
 <b>by </b><i><a class="ref" href="orders_2.html#T29" title="ORDERS_2:th.29">ORDERS_2:29</a></i>;<br/><a NAME="E4:9_1_1_3_5_1_4"/>
<font color="Maroon" title="c13">f1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <span class="p1">(<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="c7">i</font>,<font color="Maroon" title="c8">j</font></span>)</span> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <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="c7">i</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E6:9_1_1_3"><i><font color="Green" title="E11">A8</font></i></a>, <a class="txt" href="#E7:9_1_1_3"><i><font color="Green" title="E12">A9</font></i></a>, <a class="txt" href="#E1:9_1_1_3_5_1_4"><i><font color="Green" title="E19">A26</font></i></a>, <a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></i>;<br/><a NAME="E5:9_1_1_3_5_1_4"/><b>then </b><i><font color="Green" title="E21">A28</font></i>: 
<font color="Maroon" title="c13">f1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="msalimit.html#K1" title="MSALIMIT:func.1">bind</a> <font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">j</font>
 <b>by </b><i><a class="ref" href="msualg_3.html#T3" title="MSUALG_3:th.3">MSUALG_3:3</a></i>;<br/><a NAME="E6:9_1_1_3_5_1_4"/>
<font color="Maroon" title="c17">f2</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <span class="p1">(<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="c8">j</font>,<font color="Maroon" title="c9">k</font></span>)</span> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <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="c8">j</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E14:9_1_1_3"><i><font color="Green" title="E14">A11</font></i></a>, <a class="txt" href="#E15:9_1_1_3"><i><font color="Green" title="E15">A12</font></i></a>, <a class="txt" href="#E1:9_1_1_3_5_1_4"><i><font color="Green" title="E19">A26</font></i></a>, <a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></i>;<br/><a NAME="E7:9_1_1_3_5_1_4"/><b>then </b>
<font color="Maroon" title="c17">f2</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="msalimit.html#K1" title="MSALIMIT:func.1">bind</a> <font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c9">k</font>
 <b>by </b><i><a class="ref" href="msualg_3.html#T3" title="MSUALG_3:th.3">MSUALG_3:3</a></i>;<br/><a NAME="E8:9_1_1_3_5_1_4"/><b>then </b>
<font color="Maroon" title="c17">f2</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c13">f1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="msalimit.html#K1" title="MSALIMIT:func.1">bind</a> <font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c9">k</font>
 <b>by </b><i><a class="txt" href="#E1:9_1_1_3"><i><font color="Green" title="E9">A6</font></i></a>, <a class="txt" href="#E5:9_1_1_3_5_1_4"><i><font color="Green" title="E21">A28</font></i></a>, <a class="ref" href="msalimit.html#T1" target="_self" title="MSALIMIT:th.1">Th1</a></i>;<br/><b>hence </b><a NAME="E9:9_1_1_3_5_1_4"/>
<font color="Maroon" title="c6">A</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">k</font>,<font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">f2</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c13">f1</font>
 <b>by </b><i><a class="txt" href="#E1:9_1_1_3"><i><font color="Green" title="E9">A6</font></i></a>, <a class="txt" href="#E19:9_1_1_3"><i><font color="Green" title="E17">A18</font></i></a>, <a class="txt" href="#E3:9_1_1_3_5_1_4"><i><font color="Green" title="E20">A27</font></i></a>, <a class="ref" href="orders_2.html#T26" title="ORDERS_2:th.26">ORDERS_2:26</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E21:9_1_1_3"/>
<font color="Maroon" title="c13">f1</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c3">OAF</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c3">OAF</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c8">j</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_3_6"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_3_6_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:9_1_1_3_6_1"/>
( <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">j</font> or <font color="Maroon" title="c7">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">j</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_3_6_1_1"><b>suppose </b></a><a NAME="E1:9_1_1_3_6_1_1"/><i><font color="Green" title="E19">A29</font></i>: 
<font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c13">f1</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c3">OAF</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c3">OAF</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c8">j</font></span><br/><div class="add"><a NAME="E2:9_1_1_3_6_1_1"/><b>then </b>
<font color="Maroon" title="c6">A</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="p1">(<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="c7">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E18:9_1_1_3"><i><font color="Green" title="E16">A13</font></i></a></i>;<br/><b>hence </b><a NAME="E3:9_1_1_3_6_1_1"/>
<font color="Maroon" title="c13">f1</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c3">OAF</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c3">OAF</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c8">j</font>
 <b>by </b><i><a class="txt" href="#E1:9_1_1_3_6_1_1"><i><font color="Green" title="E19">A29</font></i></a>, <a class="ref" href="msualg_3.html#T9" title="MSUALG_3:th.9">MSUALG_3:9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_3_6_1_2"><b>suppose </b></a><a NAME="E1:9_1_1_3_6_1_2"/>
<font color="Maroon" title="c7">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c13">f1</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c3">OAF</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c3">OAF</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c8">j</font></span><br/><div class="add"><a NAME="E2:9_1_1_3_6_1_2"/><b>then </b>
<font color="Maroon" title="c6">A</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msalimit.html#K1" title="MSALIMIT:func.1">bind</a> <font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">j</font>
 <b>by </b><i><a class="txt" href="#E1:9_1_1_3"><i><font color="Green" title="E9">A6</font></i></a>, <a class="txt" href="#E19:9_1_1_3"><i><font color="Green" title="E17">A18</font></i></a></i>;<br/><b>hence </b><a NAME="E3:9_1_1_3_6_1_2"/>
<font color="Maroon" title="c13">f1</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c3">OAF</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c3">OAF</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c8">j</font>
 <b>by </b><i><a class="txt" href="#E1:9_1_1_3"><i><font color="Green" title="E9">A6</font></i></a>, <a class="ref" href="msalimit.html#T2" target="_self" title="MSALIMIT:th.2">Th2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>hence </b><a NAME="E22:9_1_1_3"/>
 ex <font color="Olive" title="b1">f1</font> being  <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <span class="p1">(<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="c7">i</font></span>)</span>,<span class="p1">(<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="c8">j</font></span>)</span> ex <font color="Olive" title="b2">f2</font> being  <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <span class="p1">(<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="c8">j</font></span>)</span>,<span class="p1">(<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="c9">k</font></span>)</span> st <br/>( <font color="Olive" title="b1">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">A</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c7">i</font> &amp; <font color="Olive" title="b2">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">A</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">k</font>,<font color="Maroon" title="c8">j</font> &amp; <font color="Maroon" title="c6">A</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">k</font>,<font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f2</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Olive" title="b1">f1</font> &amp; <font color="Olive" title="b1">f1</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c3">OAF</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c3">OAF</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c8">j</font> )
 <b>by </b><i><a class="txt" href="#E20:9_1_1_3"><i><font color="Green" title="E18">A22</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c7">A</font> = <font color="Maroon" title="c6">A</font> as    <a href="msalimit.html#M2" title="MSALIMIT:mode.2">Binding</a> of <font color="Maroon" title="c3">OAF</font> <b>by </b><i><a class="ref" href="msalimit.html#D2" target="_self" title="MSALIMIT:def.2">Def2</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c7">A</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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="c7">A</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#K14" title="FUNCOP_1:func.14">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><br/>

<b>let </b><font color="Maroon" title="c8">i</font>, <font color="Maroon" title="c9">j</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">P</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">i</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c9">j</font> implies <font color="Maroon" title="c7">A</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">j</font>,<font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">IFEQ</a> <font color="Maroon" title="c9">j</font>,<font color="Maroon" title="c8">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="Maroon" title="c8">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="Maroon" title="c8">i</font>,<font color="Maroon" title="c9">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="Maroon" title="c8">i</font></span>)</span></span>)</span></span>)</span> )</span><br/>



<b>assume </b><a NAME="E9:9_1_1"/><i><font color="Green" title="E9">A30</font></i>: 
<font color="Maroon" title="c8">i</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c9">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">A</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">j</font>,<font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">IFEQ</a> <font color="Maroon" title="c9">j</font>,<font color="Maroon" title="c8">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="Maroon" title="c8">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="Maroon" title="c8">i</font>,<font color="Maroon" title="c9">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="Maroon" title="c8">i</font></span>)</span></span>)</span></span>)</span></span><br/>

<b>consider </b><font color="Maroon" title="c10">kl</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E11:9_1_1"/><i><font color="Green" title="E10">A31</font></i>: 
<font color="Maroon" title="c10">kl</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"><font color="Maroon" title="c9">j</font>,<font color="Maroon" title="c8">i</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 ;<br/>
<a NAME="E12:9_1_1"/>
<font color="Maroon" title="c10">kl</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>
 
<b>by </b><i><a class="txt" href="#E9:9_1_1"><i><font color="Green" title="E9">A30</font></i></a>, <a class="txt" href="#E11:9_1_1"><i><font color="Green" title="E10">A31</font></i></a>, <a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c11">i1</font>, <font color="Maroon" title="c12">j1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">P</font><b> such that </b><br/><a NAME="E14:9_1_1"/><i><font color="Green" title="E11">A32</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">j1</font>,<font color="Maroon" title="c11">i1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">kl</font> &amp; <font color="Maroon" title="c7">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">kl</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">IFEQ</a> <font color="Maroon" title="c12">j1</font>,<font color="Maroon" title="c11">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="c11">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="c11">i1</font>,<font color="Maroon" title="c12">j1</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="c11">i1</font></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="#E4:9_1_1"><i><font color="Green" title="E8">A4</font></i></a></i>;<br/>
<a NAME="E15:9_1_1"/>
( <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">i</font> &amp; <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">j</font> )
 
<b>by </b><i><a class="txt" href="#E11:9_1_1"><i><font color="Green" title="E10">A31</font></i></a>, <a class="txt" href="#E14:9_1_1"><i><font color="Green" title="E11">A32</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E16:9_1_1"/>
<font color="Maroon" title="c7">A</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">j</font>,<font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">IFEQ</a> <font color="Maroon" title="c9">j</font>,<font color="Maroon" title="c8">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="Maroon" title="c8">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="Maroon" title="c8">i</font>,<font color="Maroon" title="c9">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="Maroon" title="c8">i</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E14:9_1_1"><i><font color="Green" title="E11">A32</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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