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<b>let </b><font color="Maroon" title="c1">f</font>, <font color="Maroon" title="c2">g</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="scmpds_1.html#K1" title="SCMPDS_1:func.1">SCMPDS-Instr</a> , <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <a href="scmpds_1.html#K2" title="SCMPDS_1:func.2">SCMPDS-OK</a> </span>)</span>,<span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <a href="scmpds_1.html#K2" title="SCMPDS_1:func.2">SCMPDS-OK</a> </span>)</span>; <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">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="scmpds_1.html#K1" title="SCMPDS_1:func.1">SCMPDS-Instr</a> <br/> for <font color="Olive" title="b2">y</font> being  <a href="scmpds_1.html#NM1" title="SCMPDS_1:NM.1">SCMPDS-State</a> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmpds_1.html#K24" title="SCMPDS_1:func.24">SCM-Exec-Res</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> ) &amp; ( for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="scmpds_1.html#K1" title="SCMPDS_1:func.1">SCMPDS-Instr</a> <br/> for <font color="Olive" title="b2">y</font> being  <a href="scmpds_1.html#NM1" title="SCMPDS_1:NM.1">SCMPDS-State</a> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmpds_1.html#K24" title="SCMPDS_1:func.24">SCM-Exec-Res</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> ) implies <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:47_1_2"/><i><font color="Green" title="E24">A2</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="scmpds_1.html#K1" title="SCMPDS_1:func.1">SCMPDS-Instr</a> <br/> for <font color="Olive" title="b2">y</font> being  <a href="scmpds_1.html#NM1" title="SCMPDS_1:NM.1">SCMPDS-State</a> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmpds_1.html#K24" title="SCMPDS_1:func.24">SCM-Exec-Res</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>
 <b>and </b><br/><a NAME="E2:47_1_2"/><i><font color="Green" title="E25">A3</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="scmpds_1.html#K1" title="SCMPDS_1:func.1">SCMPDS-Instr</a> <br/> for <font color="Olive" title="b2">y</font> being  <a href="scmpds_1.html#NM1" title="SCMPDS_1:NM.1">SCMPDS-State</a> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmpds_1.html#K24" title="SCMPDS_1:func.24">SCM-Exec-Res</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="scmpds_1.html#K1" title="SCMPDS_1:func.1">SCMPDS-Instr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">f</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c3">x</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c2">g</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c3">x</font></span><br/><b>reconsider </b><font color="Maroon" title="c4">gx</font> = <font color="Maroon" title="c2">g</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c5">fx</font> = <font color="Maroon" title="c1">f</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c3">x</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <a href="scmpds_1.html#K2" title="SCMPDS_1:func.2">SCMPDS-OK</a> , <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <a href="scmpds_1.html#K2" title="SCMPDS_1:func.2">SCMPDS-OK</a>  ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">y</font> be   <a href="scmpds_1.html#NM1" title="SCMPDS_1:NM.1">SCMPDS-State</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">fx</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">gx</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">y</font></span><br/><b>thus </b><font color="Maroon" title="c5">fx</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">y</font> = 
 <a href="scmpds_1.html#K24" title="SCMPDS_1:func.24">SCM-Exec-Res</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c6">y</font>
<b>by </b><i><a class="txt" href="scmpds_1.html#E1:47_1_2"><i><font color="Green" title="E24">A2</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c4">gx</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">y</font>
<b>by </b><i><a class="txt" href="scmpds_1.html#E2:47_1_2"><i><font color="Green" title="E25">A3</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="E3:47_1_2_1"/>
<font color="Maroon" title="c1">f</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c3">x</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c2">g</font> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c3">x</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</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="E4:47_1_2"/>
<font color="Maroon" title="c1">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c2">g</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</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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