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<b>let </b><font color="Maroon" title="c3">f1</font>, <font color="Maroon" title="c4">f2</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"> (  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c1">P</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; ( for <font color="Olive" title="b1">m</font> being   <a href="pnproc_1.html#M2" title="PNPROC_1:mode.2">marking</a> of <font color="Maroon" title="c1">P</font> holds  <font color="Maroon" title="c3">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pnproc_1.html#K6" title="PNPROC_1:func.6">fire</a> <font color="Maroon" title="c2">t</font>,<font color="Olive" title="b1">m</font> ) &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c1">P</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; ( for <font color="Olive" title="b1">m</font> being   <a href="pnproc_1.html#M2" title="PNPROC_1:mode.2">marking</a> of <font color="Maroon" title="c1">P</font> holds  <font color="Maroon" title="c4">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pnproc_1.html#K6" title="PNPROC_1:func.6">fire</a> <font color="Maroon" title="c2">t</font>,<font color="Olive" title="b1">m</font> ) implies <font color="Maroon" title="c3">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f2</font> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:54_1_2"/><i><font color="Green" title="E37">A5</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c1">P</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>and </b><br/><a NAME="E2:54_1_2"/><i><font color="Green" title="E38">A6</font></i>: 
for <font color="Olive" title="b1">m</font> being   <a href="pnproc_1.html#M2" title="PNPROC_1:mode.2">marking</a> of <font color="Maroon" title="c1">P</font> holds  <font color="Maroon" title="c3">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pnproc_1.html#K6" title="PNPROC_1:func.6">fire</a> <font color="Maroon" title="c2">t</font>,<font color="Olive" title="b1">m</font>
 <b>and </b><br/><a NAME="E3:54_1_2"/><i><font color="Green" title="E39">A7</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c1">P</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>and </b><br/><a NAME="E4:54_1_2"/><i><font color="Green" title="E40">A8</font></i>: 
for <font color="Olive" title="b1">m</font> being   <a href="pnproc_1.html#M2" title="PNPROC_1:mode.2">marking</a> of <font color="Maroon" title="c1">P</font> holds  <font color="Maroon" title="c4">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pnproc_1.html#K6" title="PNPROC_1:func.6">fire</a> <font color="Maroon" title="c2">t</font>,<font color="Olive" title="b1">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f2</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">x</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">x</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> <font color="Maroon" title="c3">f1</font> implies <font color="Maroon" title="c3">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> )</span><br/><b>assume </b><a NAME="E1:54_1_2_1"/>
<font color="Maroon" title="c5">x</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> <font color="Maroon" title="c3">f1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font></span><br/><b>then reconsider </b><font color="Maroon" title="c6">m</font> = <font color="Maroon" title="c5">x</font> as    <a href="pnproc_1.html#M2" title="PNPROC_1:mode.2">marking</a> of <font color="Maroon" title="c1">P</font> <b>by </b><i><a class="txt" href="pnproc_1.html#E1:54_1_2"><i><font color="Green" title="E37">A5</font></i></a>, <a class="ref" href="pnproc_1.html#T32" target="_self" title="PNPROC_1:th.32">Th32</a></i>;<br/><b>thus </b><font color="Maroon" title="c3">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> = 
 <a href="pnproc_1.html#K6" title="PNPROC_1:func.6">fire</a> <font color="Maroon" title="c2">t</font>,<font color="Maroon" title="c6">m</font>
<b>by </b><i><a class="txt" href="pnproc_1.html#E2:54_1_2"><i><font color="Green" title="E38">A6</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c4">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font>
<b>by </b><i><a class="txt" href="pnproc_1.html#E4:54_1_2"><i><font color="Green" title="E40">A8</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="E6:54_1_2"/>
<font color="Maroon" title="c3">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f2</font>
 <b>by </b><i><a class="txt" href="pnproc_1.html#E1:54_1_2"><i><font color="Green" title="E37">A5</font></i></a>, <a class="txt" href="pnproc_1.html#E3:54_1_2"><i><font color="Green" title="E39">A7</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</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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