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<b>let </b><font color="Maroon" title="c7">ti</font> be    <a href="aofa_i00.html#M3" title="AOFA_I00:mode.3">INT-Expression</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c4">f</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">ti</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aofa_i00.html#K5" title="AOFA_I00:func.5">gt</a> <font color="Maroon" title="c5">t1</font>,<font color="Maroon" title="c6">t2</font> iff for <font color="Olive" title="b1">s</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  holds  <font color="Maroon" title="c7">ti</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K5" title="XXREAL_0:func.5">IFGT</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">t1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">t2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font></span>)</span>,1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/>

<a NAME="E1:59_4_1"/><i><font color="Green" title="E35">A1</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="aofa_i00.html#K13" title="AOFA_I00:func.13">gt</a> <font color="Maroon" title="c5">t1</font>,<font color="Maroon" title="c6">t2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">t1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">t2</font></span>)</span> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">t2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">ti</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  )
 
<b>by </b><i><a class="ref" href="aofa_i00.html#D6" target="_self" title="AOFA_I00:def.6">DEFgt</a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E2:59_4_1"/>
( <font color="Maroon" title="c7">ti</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="aofa_i00.html#K13" title="AOFA_I00:func.13">gt</a> <font color="Maroon" title="c5">t1</font>,<font color="Maroon" title="c6">t2</font> implies for <font color="Olive" title="b1">s</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  holds  <font color="Maroon" title="c7">ti</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K5" title="XXREAL_0:func.5">IFGT</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">t1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">t2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font></span>)</span>,1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="ref" href="aofa_i00.html#D6" target="_self" title="AOFA_I00:def.6">DEFgt</a></i>; <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">s</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  holds  <font color="Maroon" title="c7">ti</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K5" title="XXREAL_0:func.5">IFGT</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">t1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">t2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font></span>)</span>,1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) implies <font color="Maroon" title="c7">ti</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aofa_i00.html#K5" title="AOFA_I00:func.5">gt</a> <font color="Maroon" title="c5">t1</font>,<font color="Maroon" title="c6">t2</font> )</span><br/>

<b>assume </b><a NAME="E3:59_4_1"/>
for <font color="Olive" title="b1">s</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  holds  <font color="Maroon" title="c7">ti</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K5" title="XXREAL_0:func.5">IFGT</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">t1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">t2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font></span>)</span>,1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">ti</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aofa_i00.html#K5" title="AOFA_I00:func.5">gt</a> <font color="Maroon" title="c5">t1</font>,<font color="Maroon" title="c6">t2</font></span><br/>

<a NAME="E4:59_4_1"/><b>then </b>
for <font color="Olive" title="b1">s</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  holds <br/><font color="Maroon" title="c7">ti</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K5" title="XXREAL_0:func.5">IFGT</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">t1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">t2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s</font></span>)</span>,1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
;<br/>
<b>hence </b><a NAME="E5:59_4_1"/>
<font color="Maroon" title="c7">ti</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="aofa_i00.html#K13" title="AOFA_I00:func.13">gt</a> <font color="Maroon" title="c5">t1</font>,<font color="Maroon" title="c6">t2</font>
 <b>by </b><i><a class="txt" href="#E1:59_4_1"><i><font color="Green" title="E35">A1</font></i></a>, <a class="ref" href="aofa_i00.html#D6" target="_self" title="AOFA_I00:def.6">DEFgt</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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