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<span class="kw">let </span><font color="Maroon" title="c1">P</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">m1</font>, <font color="Olive" title="b2">m2</font> being  <a href="pnproc_1.html#NM1" title="PNPROC_1:NM.1">marking</a> of <font color="Maroon" title="c1">P</font>  st <font color="Olive" title="b1">m1</font> <a href="pnproc_1.html#R2" title="PNPROC_1:pred.2">c=</a> <font color="Olive" title="b2">m2</font> holds <br/><font color="Olive" title="b2">m2</font> <a href="pnproc_1.html#K3" title="PNPROC_1:func.3">-</a> <font color="Olive" title="b1">m1</font> <a href="pnproc_1.html#R2" title="PNPROC_1:pred.2">c=</a> <font color="Olive" title="b2">m2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">m1</font>, <font color="Maroon" title="c3">m2</font> be   <a href="pnproc_1.html#NM1" title="PNPROC_1:NM.1">marking</a> of <font color="Maroon" title="c1">P</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">m1</font> <a href="pnproc_1.html#R2" title="PNPROC_1:pred.2">c=</a> <font color="Maroon" title="c3">m2</font> implies <font color="Maroon" title="c3">m2</font> <a href="pnproc_1.html#K3" title="PNPROC_1:func.3">-</a> <font color="Maroon" title="c2">m1</font> <a href="pnproc_1.html#R2" title="PNPROC_1:pred.2">c=</a> <font color="Maroon" title="c3">m2</font> )</span><br/>





<span class="kw">assume </span><a NAME="E1:40"/><span class="lab"><font color="Green" title="E27">A1</font></span>: 
<font color="Maroon" title="c2">m1</font> <a href="pnproc_1.html#R2" title="PNPROC_1:pred.2">c=</a> <font color="Maroon" title="c3">m2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">m2</font> <a href="pnproc_1.html#K3" title="PNPROC_1:func.3">-</a> <font color="Maroon" title="c2">m1</font> <a href="pnproc_1.html#R2" title="PNPROC_1:pred.2">c=</a> <font color="Maroon" title="c3">m2</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c4">p</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="pnproc_1.html#D4" target="_self" title="PNPROC_1:def.4">PNPROC_1:def 4</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">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="c4">p</font> <a href="pnproc_1.html#NK1" title="PNPROC_1:NK.1">multitude_of</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">p</font> <a href="pnproc_1.html#NK1" title="PNPROC_1:NK.1">multitude_of</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E2:40"/>
<font color="Maroon" title="c4">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">p</font> <a href="pnproc_1.html#NK1" title="PNPROC_1:NK.1">multitude_of</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">p</font> <a href="pnproc_1.html#NK1" title="PNPROC_1:NK.1">multitude_of</a> </span><br/>

<a NAME="E3:40"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A2</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">m2</font> <a href="pnproc_1.html#K3" title="PNPROC_1:func.3">-</a> <font color="Maroon" title="c2">m1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">p</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="ref" href="pnproc_1.html#D6" target="_self" title="PNPROC_1:def.6">Def6</a></span>;<br/>
<a NAME="E4:40"/><span class="lab"><font color="Green" title="E29">A3</font></span>: 
<font color="Maroon" title="c4">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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</a></span>;<br/>
<a NAME="E5:40"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">m2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">p</font>
 
;<br/>
<span class="kw">hence </span><a NAME="E6:40"/>
<font color="Maroon" title="c4">p</font> <a href="pnproc_1.html#NK1" title="PNPROC_1:NK.1">multitude_of</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">p</font> <a href="pnproc_1.html#NK1" title="PNPROC_1:NK.1">multitude_of</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:40"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="txt" href="#E4:40"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="ref" href="xreal_1.html#T15" title="XREAL_1:th.15">XREAL_1:15</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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