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<b>consider </b><font color="Maroon">c<sub>2</sub></font> being  non <a href="xboole_0.html#V1">empty</a> <a href="unialg_1.html#V1">homogeneous</a> <a href="unialg_1.html#V2">quasi_total</a> <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">c<sub>1</sub></font> <a href="finseq_2.html#K3">*</a> ,<font color="Maroon">c<sub>1</sub></font>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="partfun1.html#K4">PFuncs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_2.html#K3">*</a> </span>)</span>,<font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="partfun1.html#T119">PARTFUN1:119</a></i>;<br/>
<b>take </b>
<span class="p1"><a href="unialg_1.html#K1">&lt;*</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="unialg_1.html#K1">*&gt;</a></span>
;<br/>

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="unialg_1.html#D4">UNIALG_1:def 4</a>
<div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">c<sub>1</sub></font> <a href="finseq_2.html#K3">*</a> ,<font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><a NAME="E1:5_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="unialg_1.html#K1">&lt;*</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="unialg_1.html#K1">*&gt;</a></span>
 ;<br/><a NAME="E2:5_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#T55">FINSEQ_1:55</a></i>;<br/><a NAME="E3:5_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E4:5_1_1_1"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="unialg_1.html#K1">&lt;*</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="unialg_1.html#K1">*&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> implies <font color="Maroon">c<sub>5</sub></font> is <a href="unialg_1.html#V1">homogeneous</a> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/></div>
<b>end;</b></div>

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="unialg_1.html#D5">UNIALG_1:def 5</a>
<div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">c<sub>1</sub></font> <a href="finseq_2.html#K3">*</a> ,<font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><a NAME="E1:5_1_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="unialg_1.html#K1">&lt;*</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="unialg_1.html#K1">*&gt;</a></span>
 ;<br/><a NAME="E2:5_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#T55">FINSEQ_1:55</a></i>;<br/><a NAME="E3:5_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E4:5_1_1_2"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="unialg_1.html#K1">&lt;*</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="unialg_1.html#K1">*&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> implies <font color="Maroon">c<sub>5</sub></font> is <a href="unialg_1.html#V2">quasi_total</a> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/></div>
<b>end;</b></div>

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D15">FUNCT_1:def 15</a>
<div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:5_1_1_3"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="unialg_1.html#K1">&lt;*</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="unialg_1.html#K1">*&gt;</a></span>
 ;<br/><a NAME="E2:5_1_1_3"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#T55">FINSEQ_1:55</a></i>;<br/><a NAME="E3:5_1_1_3"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E4:5_1_1_3"/>
not <span class="p1"><a href="unialg_1.html#K1">&lt;*</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="unialg_1.html#K1">*&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/></div>
<b>end;</b></div>

<b>thus </b><a NAME="E6:5_1_1"/>
not <span class="p1"><a href="unialg_1.html#K1">&lt;*</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="unialg_1.html#K1">*&gt;</a></span> is <a href="xboole_0.html#V1">empty</a>
 ;<br/>


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