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<b>let </b><font color="Maroon" title="c1">U1</font>, <font color="Maroon" title="c2">U2</font>, <font color="Maroon" title="c3">U3</font> be   <a href="unialg_1.html#NM2" title="UNIALG_1:NM.2">Universal_Algebra</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">U2</font> <a href="unialg_2.html#R1" title="UNIALG_2:pred.1">are_similar</a>  &amp; <font color="Maroon" title="c2">U2</font>,<font color="Maroon" title="c3">U3</font> <a href="unialg_2.html#R1" title="UNIALG_2:pred.1">are_similar</a>  implies for <font color="Olive" title="b1">h1</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">U2</font><br/> for <font color="Olive" title="b2">h2</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">U2</font>,<font color="Maroon" title="c3">U3</font> holds  <span class="p1">(<span class="default"><a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <font color="Olive" title="b2">h2</font></span>)</span> <a href="pboole.html#K11" title="PBOOLE:func.11">**</a> <span class="p1">(<span class="default"><a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <font color="Olive" title="b1">h1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">h2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b1">h1</font></span>)</span> )</span><br/>





<b>assume </b><b>that </b><br/><a NAME="E1:33"/><i><font color="Green" title="E25">A1</font></i>: 
<font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">U2</font> <a href="unialg_2.html#R1" title="UNIALG_2:pred.1">are_similar</a> 
 <b>and </b><br/><a NAME="E2:33"/><i><font color="Green" title="E26">A2</font></i>: 
<font color="Maroon" title="c2">U2</font>,<font color="Maroon" title="c3">U3</font> <a href="unialg_2.html#R1" title="UNIALG_2:pred.1">are_similar</a> 
 ; <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">h1</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">U2</font><br/> for <font color="Olive" title="b2">h2</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">U2</font>,<font color="Maroon" title="c3">U3</font> holds  <span class="p1">(<span class="default"><a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <font color="Olive" title="b2">h2</font></span>)</span> <a href="pboole.html#K11" title="PBOOLE:func.11">**</a> <span class="p1">(<span class="default"><a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <font color="Olive" title="b1">h1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">h2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b1">h1</font></span>)</span></span><br/>

<b>let </b><font color="Maroon" title="c4">h1</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">U2</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">h2</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">U2</font>,<font color="Maroon" title="c3">U3</font> holds  <span class="p1">(<span class="default"><a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <font color="Olive" title="b1">h2</font></span>)</span> <a href="pboole.html#K11" title="PBOOLE:func.11">**</a> <span class="p1">(<span class="default"><a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">h2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">h1</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c5">h2</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">U2</font>,<font color="Maroon" title="c3">U3</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <font color="Maroon" title="c5">h2</font></span>)</span> <a href="pboole.html#K11" title="PBOOLE:func.11">**</a> <span class="p1">(<span class="default"><a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">h2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">h1</font></span>)</span></span><br/>



<a NAME="E3:33"/><i><font color="Green" title="E27">A3</font></i>: 
 <a href="msualg_1.html#K7" title="MSUALG_1:func.7">MSSign</a> <font color="Maroon" title="c2">U2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_1.html#K7" title="MSUALG_1:func.7">MSSign</a> <font color="Maroon" title="c3">U3</font>
 
<b>by </b><i><a class="txt" href="msuhom_1.html#E2:33"><i><font color="Green" title="E26">A2</font></i></a>, <a class="ref" href="msuhom_1.html#T10" target="_self" title="MSUHOM_1:th.10">Th10</a></i>;<br/>
<a NAME="E4:33"/><i><font color="Green" title="E28">A4</font></i>: 
 <a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <font color="Maroon" title="c4">h1</font> is    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="msuhom_1.html#E1:33"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="msuhom_1.html#T17" target="_self" title="MSUHOM_1:th.17">Th17</a></i>;<br/>
<a NAME="E5:33"/>
 <a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <font color="Maroon" title="c5">h2</font> is    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="msuhom_1.html#E2:33"><i><font color="Green" title="E26">A2</font></i></a>, <a class="ref" href="msuhom_1.html#T17" target="_self" title="MSUHOM_1:th.17">Th17</a></i>;<br/>
<a NAME="E6:33"/><b>then </b><i><font color="Green" title="E29">A5</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <font color="Maroon" title="c5">h2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<i><font color="Green" title="E30">A6</font></i>:  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <font color="Maroon" title="c5">h2</font></span>)</span> <a href="pboole.html#K11" title="PBOOLE:func.11">**</a> <span class="p2">(<span class="default"><a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <font color="Maroon" title="c4">h1</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <font color="Maroon" title="c4">h1</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <font color="Maroon" title="c5">h2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="pboole.html#D24" title="PBOOLE:def.24">PBOOLE:def 24</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<b>by </b><i><a class="txt" href="msuhom_1.html#E4:33"><i><font color="Green" title="E28">A4</font></i></a>, <a class="txt" href="msuhom_1.html#E6:33"><i><font color="Green" title="E29">A5</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>

;<br/>
<div><i><font color="Green" title="E31">A7</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="33_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">a</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"> for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Maroon" title="c6">a</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <font color="Maroon" title="c5">h2</font></span>)</span> <a href="pboole.html#K11" title="PBOOLE:func.11">**</a> <span class="p2">(<span class="default"><a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <font color="Maroon" title="c4">h1</font></span>)</span></span>)</span> &amp; <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">a</font> &amp; <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <font color="Maroon" title="c5">h2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">a</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <font color="Maroon" title="c5">h2</font></span>)</span> <a href="pboole.html#K11" title="PBOOLE:func.11">**</a> <span class="p2">(<span class="default"><a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <font color="Maroon" title="c4">h1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">h1</font></span><br/><b>let </b><font color="Maroon" title="c7">f</font>, <font color="Maroon" title="c8">g</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</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">a</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <font color="Maroon" title="c5">h2</font></span>)</span> <a href="pboole.html#K11" title="PBOOLE:func.11">**</a> <span class="p2">(<span class="default"><a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <font color="Maroon" title="c4">h1</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c7">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">a</font> &amp; <font color="Maroon" title="c8">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <font color="Maroon" title="c5">h2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">a</font> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <font color="Maroon" title="c5">h2</font></span>)</span> <a href="pboole.html#K11" title="PBOOLE:func.11">**</a> <span class="p2">(<span class="default"><a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <font color="Maroon" title="c4">h1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">h1</font> )</span><br/><b>assume </b><b>that </b><br/><a NAME="E1:33_2"/><i><font color="Green" title="E32">A8</font></i>: 
<font color="Maroon" title="c6">a</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <font color="Maroon" title="c5">h2</font></span>)</span> <a href="pboole.html#K11" title="PBOOLE:func.11">**</a> <span class="p2">(<span class="default"><a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <font color="Maroon" title="c4">h1</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E2:33_2"/><i><font color="Green" title="E33">A9</font></i>: 
<font color="Maroon" title="c7">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">a</font>
 <b>and </b><br/><a NAME="E3:33_2"/><i><font color="Green" title="E34">A10</font></i>: 
<font color="Maroon" title="c8">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <font color="Maroon" title="c5">h2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <font color="Maroon" title="c5">h2</font></span>)</span> <a href="pboole.html#K11" title="PBOOLE:func.11">**</a> <span class="p2">(<span class="default"><a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <font color="Maroon" title="c4">h1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">h1</font></span><br/><i><font color="Green" title="E35">A11</font></i>: <font color="Maroon" title="c7">f</font> = 
<span class="p1">(<span class="default"><a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="txt" href="msuhom_1.html#E7:33"><i><font color="Green" title="E30">A6</font></i></a>, <a class="txt" href="msuhom_1.html#E1:33_2"><i><font color="Green" title="E32">A8</font></i></a>, <a class="txt" href="msuhom_1.html#E2:33_2"><i><font color="Green" title="E33">A9</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="txt" href="msuhom_1.html#E1:33"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="msuhom_1.html#D3" target="_self" title="MSUHOM_1:def.3">Def3</a>, <a class="ref" href="msuhom_1.html#T10" target="_self" title="MSUHOM_1:th.10">Th10</a></i>
<br/>.= 
<font color="Maroon" title="c4">h1</font>
<b>by </b><i><a class="ref" href="funcop_1.html#T87" title="FUNCOP_1:th.87">FUNCOP_1:87</a></i>
;<br/><font color="Maroon" title="c8">g</font> = 
<span class="p1">(<span class="default"><a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <font color="Maroon" title="c5">h2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="txt" href="msuhom_1.html#E7:33"><i><font color="Green" title="E30">A6</font></i></a>, <a class="txt" href="msuhom_1.html#E1:33_2"><i><font color="Green" title="E32">A8</font></i></a>, <a class="txt" href="msuhom_1.html#E3:33_2"><i><font color="Green" title="E34">A10</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Maroon" title="c5">h2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="txt" href="msuhom_1.html#E2:33"><i><font color="Green" title="E26">A2</font></i></a>, <a class="ref" href="msuhom_1.html#D3" target="_self" title="MSUHOM_1:def.3">Def3</a>, <a class="ref" href="msuhom_1.html#T10" target="_self" title="MSUHOM_1:th.10">Th10</a></i>
<br/>.= 
<font color="Maroon" title="c5">h2</font>
<b>by </b><i><a class="ref" href="funcop_1.html#T87" title="FUNCOP_1:th.87">FUNCOP_1:87</a></i>
;<br/><b>hence </b><a NAME="E6:33_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <font color="Maroon" title="c5">h2</font></span>)</span> <a href="pboole.html#K11" title="PBOOLE:func.11">**</a> <span class="p2">(<span class="default"><a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <font color="Maroon" title="c4">h1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">h1</font>
 <b>by </b><i><a class="txt" href="msuhom_1.html#E1:33_2"><i><font color="Green" title="E32">A8</font></i></a>, <a class="txt" href="msuhom_1.html#E2:33_2"><i><font color="Green" title="E33">A9</font></i></a>, <a class="txt" href="msuhom_1.html#E3:33_2"><i><font color="Green" title="E34">A10</font></i></a>, <a class="txt" href="msuhom_1.html#E4:33_2"><i><font color="Green" title="E35">A11</font></i></a>, <a class="ref" href="pboole.html#D24" title="PBOOLE:def.24">PBOOLE:def 24</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="E9:33"/><i><font color="Green" title="E32">A12</font></i>: 
<span class="p1">(<span class="default"><a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <font color="Maroon" title="c5">h2</font></span>)</span> <a href="pboole.html#K11" title="PBOOLE:func.11">**</a> <span class="p1">(<span class="default"><a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <font color="Maroon" title="c4">h1</font></span>)</span> is    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="msuhom_1.html#E7:33"><i><font color="Green" title="E30">A6</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<b>set </b><font color="Maroon" title="c6">h</font> = <font color="Maroon" title="c5">h2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">h1</font>;<br/>
<a NAME="E10:33"/><i><font color="Green" title="E33">A13</font></i>: 
 <a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">h2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">h1</font></span>)</span> is    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="msuhom_1.html#E1:33"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="msuhom_1.html#E2:33"><i><font color="Green" title="E26">A2</font></i></a>, <a class="ref" href="msuhom_1.html#T17" target="_self" title="MSUHOM_1:th.17">Th17</a>, <a class="ref" href="unialg_2.html#T4" title="UNIALG_2:th.4">UNIALG_2:4</a></i>;<br/>
<a NAME="E11:33"/><b>then </b><i><font color="Green" title="E34">A14</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">h2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">h1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<div><i><font color="Green" title="E35">A15</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="33_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">a</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="c7">a</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> <span class="p1">(<span class="default"><a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">h2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">h1</font></span>)</span></span>)</span> implies <span class="p1">(<span class="default"><a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">h2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">h1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">h1</font> )</span><br/><b>assume </b><a NAME="E1:33_3"/>
<font color="Maroon" title="c7">a</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> <span class="p1">(<span class="default"><a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">h2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">h1</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">h2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">h1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">h1</font></span><br/><a NAME="E2:33_3"/><b>then </b><i><font color="Green" title="E36">A16</font></i>: 
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="msuhom_1.html#E10:33"><i><font color="Green" title="E33">A13</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><span class="p1">(<span class="default"><a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">h2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">h1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  = 
<span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">h2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">h1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="txt" href="msuhom_1.html#E1:33"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="msuhom_1.html#E3:33"><i><font color="Green" title="E27">A3</font></i></a>, <a class="ref" href="msuhom_1.html#D3" target="_self" title="MSUHOM_1:def.3">Def3</a>, <a class="ref" href="msuhom_1.html#T10" target="_self" title="MSUHOM_1:th.10">Th10</a></i>
<br/>.= 
<font color="Maroon" title="c5">h2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">h1</font>
<b>by </b><i><a class="ref" href="funcop_1.html#T87" title="FUNCOP_1:th.87">FUNCOP_1:87</a></i>
;<br/><b>hence </b><a NAME="E4:33_3"/>
<span class="p1">(<span class="default"><a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">h2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">h1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">h1</font>
 <b>by </b><i><a class="txt" href="msuhom_1.html#E2:33_3"><i><font color="Green" title="E36">A16</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</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="E13:33"/><i><font color="Green" title="E36">A17</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">h2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">h1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="msuhom_1.html#E10:33"><i><font color="Green" title="E33">A13</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<div><i><font color="Green" title="E37">A18</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="33_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">a</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"> for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Maroon" title="c7">a</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> <span class="p1">(<span class="default"><a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">h2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">h1</font></span>)</span></span>)</span> &amp; <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">a</font> &amp; <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <font color="Maroon" title="c5">h2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">a</font> holds <br/><span class="p1">(<span class="default"><a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">h2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">h1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <font color="Maroon" title="c5">h2</font></span>)</span> <a href="pboole.html#K11" title="PBOOLE:func.11">**</a> <span class="p2">(<span class="default"><a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <font color="Maroon" title="c4">h1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">a</font></span><br/><b>let </b><font color="Maroon" title="c8">f</font>, <font color="Maroon" title="c9">g</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>; <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="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">h2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">h1</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">a</font> &amp; <font color="Maroon" title="c9">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <font color="Maroon" title="c5">h2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">a</font> implies <span class="p1">(<span class="default"><a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">h2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">h1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <font color="Maroon" title="c5">h2</font></span>)</span> <a href="pboole.html#K11" title="PBOOLE:func.11">**</a> <span class="p2">(<span class="default"><a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <font color="Maroon" title="c4">h1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">a</font> )</span><br/><b>assume </b><b>that </b><br/><a NAME="E1:33_4"/><i><font color="Green" title="E38">A19</font></i>: 
<font color="Maroon" title="c7">a</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> <span class="p1">(<span class="default"><a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">h2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">h1</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E2:33_4"/><i><font color="Green" title="E39">A20</font></i>: 
<font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">a</font>
 <b>and </b><br/><a NAME="E3:33_4"/><i><font color="Green" title="E40">A21</font></i>: 
<font color="Maroon" title="c9">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <font color="Maroon" title="c5">h2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <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"> <span class="p1">(<span class="default"><a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">h2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">h1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <font color="Maroon" title="c5">h2</font></span>)</span> <a href="pboole.html#K11" title="PBOOLE:func.11">**</a> <span class="p2">(<span class="default"><a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <font color="Maroon" title="c4">h1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">a</font></span><br/><b>thus </b><span class="p1">(<span class="default"><a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">h2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">h1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">a</font> = 
<font color="Maroon" title="c5">h2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">h1</font>
<b>by </b><i><a class="txt" href="msuhom_1.html#E12:33"><i><font color="Green" title="E35">A15</font></i></a>, <a class="txt" href="msuhom_1.html#E1:33_4"><i><font color="Green" title="E38">A19</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <font color="Maroon" title="c5">h2</font></span>)</span> <a href="pboole.html#K11" title="PBOOLE:func.11">**</a> <span class="p2">(<span class="default"><a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <font color="Maroon" title="c4">h1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">a</font>
<b>by </b><i><a class="txt" href="msuhom_1.html#E7:33"><i><font color="Green" title="E30">A6</font></i></a>, <a class="txt" href="msuhom_1.html#E8:33"><i><font color="Green" title="E31">A7</font></i></a>, <a class="txt" href="msuhom_1.html#E11:33"><i><font color="Green" title="E34">A14</font></i></a>, <a class="txt" href="msuhom_1.html#E1:33_4"><i><font color="Green" title="E38">A19</font></i></a>, <a class="txt" href="msuhom_1.html#E2:33_4"><i><font color="Green" title="E39">A20</font></i></a>, <a class="txt" href="msuhom_1.html#E3:33_4"><i><font color="Green" title="E40">A21</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>
<a NAME="E15:33"/>
for <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <font color="Maroon" title="c5">h2</font></span>)</span> <a href="pboole.html#K11" title="PBOOLE:func.11">**</a> <span class="p2">(<span class="default"><a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <font color="Maroon" title="c4">h1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">h2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">h1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">a</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="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <font color="Maroon" title="c5">h2</font></span>)</span> <a href="pboole.html#K11" title="PBOOLE:func.11">**</a> <span class="p2">(<span class="default"><a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <font color="Maroon" title="c4">h1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">h2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">h1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">a</font> )</span><br/>

<b>assume </b><a NAME="E1:33_5"/><i><font color="Green" title="E38">A22</font></i>: 
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <font color="Maroon" title="c5">h2</font></span>)</span> <a href="pboole.html#K11" title="PBOOLE:func.11">**</a> <span class="p2">(<span class="default"><a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <font color="Maroon" title="c4">h1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">h2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">h1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">a</font></span><br/>

<a NAME="E2:33_5"/><i><font color="Green" title="E39">A23</font></i>: 
<span class="p1">(<span class="default"><a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">a</font> is   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>
 
;<br/>
<a NAME="E3:33_5"/>
<span class="p1">(<span class="default"><a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <font color="Maroon" title="c5">h2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">a</font> is   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>
 
;<br/>
<b>hence </b><a NAME="E4:33_5"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <font color="Maroon" title="c5">h2</font></span>)</span> <a href="pboole.html#K11" title="PBOOLE:func.11">**</a> <span class="p2">(<span class="default"><a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <font color="Maroon" title="c4">h1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">h2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">h1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">a</font>
 <b>by </b><i><a class="txt" href="msuhom_1.html#E13:33"><i><font color="Green" title="E36">A17</font></i></a>, <a class="txt" href="msuhom_1.html#E14:33"><i><font color="Green" title="E37">A18</font></i></a>, <a class="txt" href="msuhom_1.html#E1:33_5"><i><font color="Green" title="E38">A22</font></i></a>, <a class="txt" href="msuhom_1.html#E2:33_5"><i><font color="Green" title="E39">A23</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>hence </b><a NAME="E16:33"/>
<span class="p1">(<span class="default"><a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <font color="Maroon" title="c5">h2</font></span>)</span> <a href="pboole.html#K11" title="PBOOLE:func.11">**</a> <span class="p1">(<span class="default"><a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">h2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">h1</font></span>)</span>
 <b>by </b><i><a class="txt" href="msuhom_1.html#E9:33"><i><font color="Green" title="E32">A12</font></i></a>, <a class="txt" href="msuhom_1.html#E10:33"><i><font color="Green" title="E33">A13</font></i></a>, <a class="ref" href="pboole.html#T3" title="PBOOLE:th.3">PBOOLE: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>
