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<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="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  = <span class="p1">(<span class="default">$1 <a href="pnproc_1.html#NK1" title="PNPROC_1:NK.1">multitude_of</a> </span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default">$1 <a href="pnproc_1.html#NK1" title="PNPROC_1:NK.1">multitude_of</a> </span>)</span>;<br/>
<b>consider </b><font color="Maroon" title="c4">m</font> being    <a href="pnproc_1.html#M2" title="PNPROC_1:mode.2">marking</a> of <font color="Maroon" title="c1">P</font><b> such that </b><br/><a NAME="E2:31_1_1"/><i><font color="Green" title="E24">A2</font></i>: 
for <font color="Olive" title="b1">p</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font> holds <br/><font color="Olive" title="b1">p</font> <a href="pnproc_1.html#NK1" title="PNPROC_1:NK.1">multitude_of</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">p</font>)
 <b>from </b><i><a class="ref" href="pnproc_1.html#S1" target="_self" title="PNPROC_1:sch.1">PNPROC_1:sch 1</a>();<br/></i>
<b>take </b>
<font color="Maroon" title="c4">m</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">p</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font> holds <br/><font color="Olive" title="b1">p</font> <a href="pnproc_1.html#NK1" title="PNPROC_1:NK.1">multitude_of</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="pnproc_1.html#NK1" title="PNPROC_1:NK.1">multitude_of</a> </span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="pnproc_1.html#NK1" title="PNPROC_1:NK.1">multitude_of</a> </span>)</span></span><br/>

<b>let </b><font color="Maroon" title="c5">p</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">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font> implies <font color="Maroon" title="c5">p</font> <a href="pnproc_1.html#NK1" title="PNPROC_1:NK.1">multitude_of</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="pnproc_1.html#NK1" title="PNPROC_1:NK.1">multitude_of</a> </span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="pnproc_1.html#NK1" title="PNPROC_1:NK.1">multitude_of</a> </span>)</span> )</span><br/>

<b>assume </b><a NAME="E3:31_1_1"/><i><font color="Green" title="E25">A3</font></i>: 
<font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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="c5">p</font> <a href="pnproc_1.html#NK1" title="PNPROC_1:NK.1">multitude_of</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="pnproc_1.html#NK1" title="PNPROC_1:NK.1">multitude_of</a> </span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="pnproc_1.html#NK1" title="PNPROC_1:NK.1">multitude_of</a> </span>)</span></span><br/>

<a NAME="E4:31_1_1"/><b>then </b><i><font color="Green" title="E26">A4</font></i>: 
<font color="Maroon" title="c5">p</font> <a href="pnproc_1.html#NK1" title="PNPROC_1:NK.1">multitude_of</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">p</font> <a href="pnproc_1.html#NK1" title="PNPROC_1:NK.1">multitude_of</a> 
 
<b>by </b><i><a class="txt" href="pnproc_1.html#E1:31"><i><font color="Green" title="E23">A1</font></i></a>, <a class="ref" href="pnproc_1.html#D4" target="_self" title="PNPROC_1:def.4">Def4</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c5">p</font> <a href="pnproc_1.html#NK1" title="PNPROC_1:NK.1">multitude_of</a>  = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="pnproc_1.html#NK1" title="PNPROC_1:NK.1">multitude_of</a> </span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="pnproc_1.html#NK1" title="PNPROC_1:NK.1">multitude_of</a> </span>)</span>
<b>by </b><i><a class="txt" href="pnproc_1.html#E2:31_1_1"><i><font color="Green" title="E24">A2</font></i></a>, <a class="txt" href="pnproc_1.html#E3:31_1_1"><i><font color="Green" title="E25">A3</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="pnproc_1.html#NK1" title="PNPROC_1:NK.1">multitude_of</a> </span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="pnproc_1.html#NK1" title="PNPROC_1:NK.1">multitude_of</a> </span>)</span>
<b>by </b><i><a class="txt" href="pnproc_1.html#E4:31_1_1"><i><font color="Green" title="E26">A4</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</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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