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<b>let </b><font color="Maroon" title="c1">F</font>, <font color="Maroon" title="c2">f</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"> for <font color="Olive" title="b1">i1</font>, <font color="Olive" title="b2">i2</font>, <font color="Olive" title="b3">xi1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b4">Ai2</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">i2</font></span>)</span>  st <font color="Olive" title="b3">xi1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i1</font> &amp; <font color="Olive" title="b1">i1</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="c1">F</font> &amp; <font color="Maroon" title="c2">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c1">F</font> &amp; <font color="Olive" title="b1">i1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">i2</font> holds <br/>( <font color="Maroon" title="c2">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K3" title="PRALG_3:func.3">proj</a> <font color="Maroon" title="c1">F</font>,<font color="Olive" title="b2">i2</font></span>)</span> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Olive" title="b4">Ai2</font> iff <font color="Maroon" title="c2">f</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Olive" title="b1">i1</font>,<font color="Olive" title="b3">xi1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K3" title="PRALG_3:func.3">proj</a> <font color="Maroon" title="c1">F</font>,<font color="Olive" title="b2">i2</font></span>)</span> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Olive" title="b4">Ai2</font> )</span><br/><b>let </b><font color="Maroon" title="c3">i1</font>, <font color="Maroon" title="c4">i2</font>, <font color="Maroon" title="c5">xi1</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"> for <font color="Olive" title="b1">Ai2</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i2</font></span>)</span>  st <font color="Maroon" title="c5">xi1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i1</font> &amp; <font color="Maroon" title="c3">i1</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="c1">F</font> &amp; <font color="Maroon" title="c2">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c1">F</font> &amp; <font color="Maroon" title="c3">i1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">i2</font> holds <br/>( <font color="Maroon" title="c2">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K3" title="PRALG_3:func.3">proj</a> <font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c4">i2</font></span>)</span> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Olive" title="b1">Ai2</font> iff <font color="Maroon" title="c2">f</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c3">i1</font>,<font color="Maroon" title="c5">xi1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K3" title="PRALG_3:func.3">proj</a> <font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c4">i2</font></span>)</span> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Olive" title="b1">Ai2</font> )</span><br/><b>let </b><font color="Maroon" title="c6">Ai2</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i2</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">xi1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i1</font> &amp; <font color="Maroon" title="c3">i1</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="c1">F</font> &amp; <font color="Maroon" title="c2">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c1">F</font> &amp; <font color="Maroon" title="c3">i1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">i2</font> implies ( <font color="Maroon" title="c2">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K3" title="PRALG_3:func.3">proj</a> <font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c4">i2</font></span>)</span> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Maroon" title="c6">Ai2</font> iff <font color="Maroon" title="c2">f</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c3">i1</font>,<font color="Maroon" title="c5">xi1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K3" title="PRALG_3:func.3">proj</a> <font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c4">i2</font></span>)</span> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Maroon" title="c6">Ai2</font> ) )</span><br/>





<b>assume </b><b>that </b><br/><a NAME="E1:7"/><i><font color="Green" title="E8">A1</font></i>: 
<font color="Maroon" title="c5">xi1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i1</font>
 <b>and </b><br/><a NAME="E2:7"/>
<font color="Maroon" title="c3">i1</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="c1">F</font>
 <b>and </b><br/><a NAME="E3:7"/><i><font color="Green" title="E9">A2</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c1">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c3">i1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">i2</font> or ( <font color="Maroon" title="c2">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K3" title="PRALG_3:func.3">proj</a> <font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c4">i2</font></span>)</span> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Maroon" title="c6">Ai2</font> iff <font color="Maroon" title="c2">f</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c3">i1</font>,<font color="Maroon" title="c5">xi1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K3" title="PRALG_3:func.3">proj</a> <font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c4">i2</font></span>)</span> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Maroon" title="c6">Ai2</font> ) )</span><br/>

<b>assume </b><a NAME="E4:7"/><i><font color="Green" title="E10">A3</font></i>: 
<font color="Maroon" title="c3">i1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">i2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K3" title="PRALG_3:func.3">proj</a> <font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c4">i2</font></span>)</span> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Maroon" title="c6">Ai2</font> iff <font color="Maroon" title="c2">f</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c3">i1</font>,<font color="Maroon" title="c5">xi1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K3" title="PRALG_3:func.3">proj</a> <font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c4">i2</font></span>)</span> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Maroon" title="c6">Ai2</font> )</span><br/>

<b>thus </b><a NAME="E5:7"/>
( <font color="Maroon" title="c2">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K3" title="PRALG_3:func.3">proj</a> <font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c4">i2</font></span>)</span> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Maroon" title="c6">Ai2</font> implies <font color="Maroon" title="c2">f</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c3">i1</font>,<font color="Maroon" title="c5">xi1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K3" title="PRALG_3:func.3">proj</a> <font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c4">i2</font></span>)</span> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Maroon" title="c6">Ai2</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">f</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c3">i1</font>,<font color="Maroon" title="c5">xi1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K3" title="PRALG_3:func.3">proj</a> <font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c4">i2</font></span>)</span> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Maroon" title="c6">Ai2</font> implies <font color="Maroon" title="c2">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K3" title="PRALG_3:func.3">proj</a> <font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c4">i2</font></span>)</span> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Maroon" title="c6">Ai2</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:7_1"/><i><font color="Green" title="E11">A4</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K3" title="PRALG_3:func.3">proj</a> <font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c4">i2</font></span>)</span> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Maroon" title="c6">Ai2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">f</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c3">i1</font>,<font color="Maroon" title="c5">xi1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K3" title="PRALG_3:func.3">proj</a> <font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c4">i2</font></span>)</span> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Maroon" title="c6">Ai2</font></span><br/>

<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c3">i1</font>,<font color="Maroon" title="c5">xi1</font></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c3">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i1</font></span>)</span> = 
<font color="Maroon" title="c2">f</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c3">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i1</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_7.html#T36" title="FUNCT_7:th.36">FUNCT_7:36</a></i>
<br/>.= 
<font color="Maroon" title="c2">f</font>
<b>by </b><i><a class="ref" href="funct_7.html#T37" title="FUNCT_7:th.37">FUNCT_7:37</a></i>
;<br/>
<b>hence </b><a NAME="E3:7_1"/>
<font color="Maroon" title="c2">f</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c3">i1</font>,<font color="Maroon" title="c5">xi1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K3" title="PRALG_3:func.3">proj</a> <font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c4">i2</font></span>)</span> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Maroon" title="c6">Ai2</font>
 <b>by </b><i><a class="txt" href="yellow17.html#E1:7"><i><font color="Green" title="E8">A1</font></i></a>, <a class="txt" href="yellow17.html#E3:7"><i><font color="Green" title="E9">A2</font></i></a>, <a class="txt" href="yellow17.html#E4:7"><i><font color="Green" title="E10">A3</font></i></a>, <a class="txt" href="yellow17.html#E1:7_1"><i><font color="Green" title="E11">A4</font></i></a>, <a class="txt" href="yellow17.html#E6"><i><font color="Green" title="E6">Lm1</font></i></a>, <a class="ref" href="yellow17.html#T2" target="_self" title="YELLOW17:th.2">Th2</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>assume </b><a NAME="E6:7"/>
<font color="Maroon" title="c2">f</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c3">i1</font>,<font color="Maroon" title="c5">xi1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K3" title="PRALG_3:func.3">proj</a> <font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c4">i2</font></span>)</span> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Maroon" title="c6">Ai2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K3" title="PRALG_3:func.3">proj</a> <font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c4">i2</font></span>)</span> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Maroon" title="c6">Ai2</font></span><br/>

<b>hence </b><a NAME="E7:7"/>
<font color="Maroon" title="c2">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K3" title="PRALG_3:func.3">proj</a> <font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c4">i2</font></span>)</span> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Maroon" title="c6">Ai2</font>
 <b>by </b><i><a class="txt" href="yellow17.html#E3:7"><i><font color="Green" title="E9">A2</font></i></a>, <a class="txt" href="yellow17.html#E4:7"><i><font color="Green" title="E10">A3</font></i></a>, <a class="txt" href="yellow17.html#E6"><i><font color="Green" title="E6">Lm1</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>
