<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="unialg_1.html#NM2">Universal_Algebra</a>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:33"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="unialg_2.html#R1">are_similar</a> 
 <b>and </b><br/><a NAME="E2:33"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="unialg_2.html#R1">are_similar</a> 
 ;<br/>

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>;<br/>



<a NAME="E3:33"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="unialg_2.html#R1">are_similar</a> 
 
<b>by </b><i><a class="txt" href="msuhom_1.html#E1:33"><i><font color="Green">E25</font></i></a>, <a class="txt" href="msuhom_1.html#E2:33"><i><font color="Green">E26</font></i></a>, <a class="ref" href="unialg_2.html#T4">UNIALG_2:4</a></i>;<br/>
<a NAME="E4:33"/><i><font color="Green">E28</font></i>: 
 <a href="msualg_1.html#K7">MSSign</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K7">MSSign</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="msuhom_1.html#E1:33"><i><font color="Green">E25</font></i></a>, <a class="ref" href="msuhom_1.html#T10" target="_self">Th10</a></i>;<br/>
<a NAME="E5:33"/><i><font color="Green">E29</font></i>: 
 <a href="msualg_1.html#K7">MSSign</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K7">MSSign</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="msuhom_1.html#E2:33"><i><font color="Green">E26</font></i></a>, <a class="ref" href="msuhom_1.html#T10" target="_self">Th10</a></i>;<br/>
<a NAME="E6:33"/><i><font color="Green">E30</font></i>: 
 <a href="msuhom_1.html#K3" target="_self">MSAlg</a> <font color="Maroon">c<sub>4</sub></font> is    <a href="pboole.html#M1">ManySortedSet</a> of <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="txt" href="msuhom_1.html#E1:33"><i><font color="Green">E25</font></i></a>, <a class="ref" href="msuhom_1.html#T17" target="_self">Th17</a></i>;<br/>
<a NAME="E7:33"/>
 <a href="msuhom_1.html#K3" target="_self">MSAlg</a> <font color="Maroon">c<sub>5</sub></font> is    <a href="pboole.html#M1">ManySortedSet</a> of <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="txt" href="msuhom_1.html#E2:33"><i><font color="Green">E26</font></i></a>, <a class="ref" href="msuhom_1.html#T17" target="_self">Th17</a></i>;<br/>
<a NAME="E8:33"/><b>then </b><i><font color="Green">E31</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="msuhom_1.html#K3" target="_self">MSAlg</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<i><font color="Green">E32</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msuhom_1.html#K3" target="_self">MSAlg</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="pboole.html#K12">**</a> <span class="p2">(<span class="default"><a href="msuhom_1.html#K3" target="_self">MSAlg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><a href="msuhom_1.html#K3" target="_self">MSAlg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><a href="msuhom_1.html#K3" target="_self">MSAlg</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="pboole.html#D24">PBOOLE:def 24</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K3">/\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="txt" href="msuhom_1.html#E6:33"><i><font color="Green">E30</font></i></a>, <a class="txt" href="msuhom_1.html#E8:33"><i><font color="Green">E31</font></i></a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>

;<br/>
<div><i><font color="Green">E33</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">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> be   <a href="funct_1.html#NM1">Function</a>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:33_2"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msuhom_1.html#K3" target="_self">MSAlg</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="pboole.html#K12">**</a> <span class="p2">(<span class="default"><a href="msuhom_1.html#K3" target="_self">MSAlg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E2:33_2"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msuhom_1.html#K3" target="_self">MSAlg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
 <b>and </b><br/><a NAME="E3:33_2"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msuhom_1.html#K3" target="_self">MSAlg</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><i><font color="Green">E37</font></i>: <font color="Maroon">c<sub>7</sub></font> = 
<span class="p1">(<span class="default"><a href="msuhom_1.html#K3" target="_self">MSAlg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> 0
<b>by </b><i><a class="txt" href="msuhom_1.html#E9:33"><i><font color="Green">E32</font></i></a>, <a class="txt" href="msuhom_1.html#E1:33_2"><i><font color="Green">E34</font></i></a>, <a class="txt" href="msuhom_1.html#E2:33_2"><i><font color="Green">E35</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> 0
<b>by </b><i><a class="txt" href="msuhom_1.html#E4:33"><i><font color="Green">E28</font></i></a>, <a class="ref" href="msuhom_1.html#D3" target="_self">Def3</a></i>
<br/>.= 
<span class="p1">(<span class="default">0 <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> 0

<br/>.= 
<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>
;<br/><font color="Maroon">c<sub>8</sub></font> = 
<span class="p1">(<span class="default"><a href="msuhom_1.html#K3" target="_self">MSAlg</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> 0
<b>by </b><i><a class="txt" href="msuhom_1.html#E9:33"><i><font color="Green">E32</font></i></a>, <a class="txt" href="msuhom_1.html#E1:33_2"><i><font color="Green">E34</font></i></a>, <a class="txt" href="msuhom_1.html#E3:33_2"><i><font color="Green">E36</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> 0
<b>by </b><i><a class="txt" href="msuhom_1.html#E5:33"><i><font color="Green">E29</font></i></a>, <a class="ref" href="msuhom_1.html#D3" target="_self">Def3</a></i>
<br/>.= 
<span class="p1">(<span class="default">0 <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> 0

<br/>.= 
<font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</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" target="_self">MSAlg</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="pboole.html#K12">**</a> <span class="p2">(<span class="default"><a href="msuhom_1.html#K3" target="_self">MSAlg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="msuhom_1.html#E1:33_2"><i><font color="Green">E34</font></i></a>, <a class="txt" href="msuhom_1.html#E2:33_2"><i><font color="Green">E35</font></i></a>, <a class="txt" href="msuhom_1.html#E3:33_2"><i><font color="Green">E36</font></i></a>, <a class="txt" href="msuhom_1.html#E4:33_2"><i><font color="Green">E37</font></i></a>, <a class="ref" href="pboole.html#D24">PBOOLE:def 24</a></i>;<br/></div><b>end;</b></div>
<a NAME="E11:33"/><i><font color="Green">E34</font></i>: 
<span class="p1">(<span class="default"><a href="msuhom_1.html#K3" target="_self">MSAlg</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="pboole.html#K12">**</a> <span class="p1">(<span class="default"><a href="msuhom_1.html#K3" target="_self">MSAlg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> is    <a href="pboole.html#M1">ManySortedSet</a> of <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="txt" href="msuhom_1.html#E9:33"><i><font color="Green">E32</font></i></a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font>;<br/>
<a NAME="E12:33"/><i><font color="Green">E35</font></i>: 
 <a href="msuhom_1.html#K3" target="_self">MSAlg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> is    <a href="pboole.html#M1">ManySortedSet</a> of <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="txt" href="msuhom_1.html#E3:33"><i><font color="Green">E27</font></i></a>, <a class="ref" href="msuhom_1.html#T17" target="_self">Th17</a></i>;<br/>
<a NAME="E13:33"/><b>then </b><i><font color="Green">E36</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="msuhom_1.html#K3" target="_self">MSAlg</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<div><i><font color="Green">E37</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">c<sub>7</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:33_3"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="msuhom_1.html#K3" target="_self">MSAlg</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
 ;<br/><a NAME="E2:33_3"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="msuhom_1.html#E12:33"><i><font color="Green">E35</font></i></a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><span class="p1">(<span class="default"><a href="msuhom_1.html#K3" target="_self">MSAlg</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> 0 = 
<span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> <a href="pre_circ.html#K2">--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> 0
<b>by </b><i><a class="txt" href="msuhom_1.html#E4:33"><i><font color="Green">E28</font></i></a>, <a class="txt" href="msuhom_1.html#E5:33"><i><font color="Green">E29</font></i></a>, <a class="ref" href="msuhom_1.html#D3" target="_self">Def3</a></i>
<br/>.= 
<span class="p1">(<span class="default">0 <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> 0

<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>
;<br/><b>hence </b><a NAME="E4:33_3"/>
<span class="p1">(<span class="default"><a href="msuhom_1.html#K3" target="_self">MSAlg</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="msuhom_1.html#E2:33_3"><i><font color="Green">E38</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div>
<a NAME="E15:33"/><i><font color="Green">E38</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="msuhom_1.html#K3" target="_self">MSAlg</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="txt" href="msuhom_1.html#E12:33"><i><font color="Green">E35</font></i></a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<div><i><font color="Green">E39</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">c<sub>7</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> be   <a href="funct_1.html#NM1">Function</a>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:33_4"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="msuhom_1.html#K3" target="_self">MSAlg</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E2:33_4"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msuhom_1.html#K3" target="_self">MSAlg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>
 <b>and </b><br/><a NAME="E3:33_4"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msuhom_1.html#K3" target="_self">MSAlg</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/><b>thus </b><span class="p1">(<span class="default"><a href="msuhom_1.html#K3" target="_self">MSAlg</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> = 
<font color="Maroon">c<sub>5</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="msuhom_1.html#E14:33"><i><font color="Green">E37</font></i></a>, <a class="txt" href="msuhom_1.html#E1:33_4"><i><font color="Green">E40</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msuhom_1.html#K3" target="_self">MSAlg</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="pboole.html#K12">**</a> <span class="p2">(<span class="default"><a href="msuhom_1.html#K3" target="_self">MSAlg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="msuhom_1.html#E9:33"><i><font color="Green">E32</font></i></a>, <a class="txt" href="msuhom_1.html#E10:33"><i><font color="Green">E33</font></i></a>, <a class="txt" href="msuhom_1.html#E13:33"><i><font color="Green">E36</font></i></a>, <a class="txt" href="msuhom_1.html#E1:33_4"><i><font color="Green">E40</font></i></a>, <a class="txt" href="msuhom_1.html#E2:33_4"><i><font color="Green">E41</font></i></a>, <a class="txt" href="msuhom_1.html#E3:33_4"><i><font color="Green">E42</font></i></a></i>
;<br/></div><b>end;</b></div>
<a NAME="E17:33"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msuhom_1.html#K3" target="_self">MSAlg</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="pboole.html#K12">**</a> <span class="p2">(<span class="default"><a href="msuhom_1.html#K3" target="_self">MSAlg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msuhom_1.html#K3" target="_self">MSAlg</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></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">c<sub>7</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:33_5"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<span class="p1">(<span class="default"><a href="msuhom_1.html#K3" target="_self">MSAlg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> 0 = 
<span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> 0
<b>by </b><i><a class="txt" href="msuhom_1.html#E4:33"><i><font color="Green">E28</font></i></a>, <a class="ref" href="msuhom_1.html#D3" target="_self">Def3</a></i>
<br/>.= 
<span class="p1">(<span class="default">0 <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> 0

<br/>.= 
<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>
;<br/>
<a NAME="E3:33_5"/><b>then </b><i><font color="Green">E41</font></i>: 
<span class="p1">(<span class="default"><a href="msuhom_1.html#K3" target="_self">MSAlg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> is   <a href="funct_1.html#NM1">Function</a>
 
<b>by </b><i><a class="txt" href="msuhom_1.html#E1:33_5"><i><font color="Green">E40</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<span class="p1">(<span class="default"><a href="msuhom_1.html#K3" target="_self">MSAlg</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> 0 = 
<span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> 0
<b>by </b><i><a class="txt" href="msuhom_1.html#E5:33"><i><font color="Green">E29</font></i></a>, <a class="ref" href="msuhom_1.html#D3" target="_self">Def3</a></i>
<br/>.= 
<span class="p1">(<span class="default">0 <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> 0

<br/>.= 
<font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>
;<br/>
<a NAME="E5:33_5"/><b>then </b>
<span class="p1">(<span class="default"><a href="msuhom_1.html#K3" target="_self">MSAlg</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> is   <a href="funct_1.html#NM1">Function</a>
 
<b>by </b><i><a class="txt" href="msuhom_1.html#E1:33_5"><i><font color="Green">E40</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E6:33_5"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msuhom_1.html#K3" target="_self">MSAlg</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="pboole.html#K12">**</a> <span class="p2">(<span class="default"><a href="msuhom_1.html#K3" target="_self">MSAlg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msuhom_1.html#K3" target="_self">MSAlg</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="msuhom_1.html#E15:33"><i><font color="Green">E38</font></i></a>, <a class="txt" href="msuhom_1.html#E16:33"><i><font color="Green">E39</font></i></a>, <a class="txt" href="msuhom_1.html#E1:33_5"><i><font color="Green">E40</font></i></a>, <a class="txt" href="msuhom_1.html#E3:33_5"><i><font color="Green">E41</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E18:33"/>
<span class="p1">(<span class="default"><a href="msuhom_1.html#K3" target="_self">MSAlg</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="pboole.html#K12">**</a> <span class="p1">(<span class="default"><a href="msuhom_1.html#K3" target="_self">MSAlg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="msuhom_1.html#K3" target="_self">MSAlg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="msuhom_1.html#E11:33"><i><font color="Green">E34</font></i></a>, <a class="txt" href="msuhom_1.html#E12:33"><i><font color="Green">E35</font></i></a>, <a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>;<br/>


</div>
