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<b>let </b><font color="Maroon" title="c10">S</font>, <font color="Maroon" title="c11">T</font> be    <a href="struct_0.html#L1" title="STRUCT_0:struct.1">1-sorted</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">f</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c10">S</font>,<font color="Maroon" title="c11">T</font>  st <font color="Maroon" title="c11">T</font> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c10">S</font> holds <br/><font color="Maroon" title="c10">S</font> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a></span><br/><b>let </b><font color="Maroon" title="c12">f</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c10">S</font>,<font color="Maroon" title="c11">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c11">T</font> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c12">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c10">S</font> implies <font color="Maroon" title="c10">S</font> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> )</span><br/>





<b>assume </b><b>that </b><br/><a NAME="E1:19"/><i><font color="Green" title="E36">A1</font></i>: 
<font color="Maroon" title="c11">T</font> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>
 <b>and </b><br/><a NAME="E2:19"/><i><font color="Green" title="E37">A2</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c12">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c10">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">S</font> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a></span><br/>

<b>assume </b><a NAME="E3:19"/>
not <font color="Maroon" title="c10">S</font> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<b>then reconsider </b><font color="Maroon" title="c13">S</font> = <font color="Maroon" title="c10">S</font> as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="struct_0.html#L1" title="STRUCT_0:struct.1">1-sorted</a>  ;<br/>
<b>consider </b><font color="Maroon" title="c14">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:19"/><i><font color="Green" title="E38">A3</font></i>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c13">S</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E7:19"/>
<font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c12">f</font>
 
<b>by </b><i><a class="txt" href="topreal2.html#E2:19"><i><font color="Green" title="E37">A2</font></i></a>, <a class="txt" href="topreal2.html#E6:19"><i><font color="Green" title="E38">A3</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E8:19"/>
contradiction
 <b>by </b><i><a class="txt" href="topreal2.html#E1:19"><i><font color="Green" title="E36">A1</font></i></a>, <a class="ref" href="struct_0.html#D1" title="STRUCT_0:def.1">STRUCT_0:def 1</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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