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<span class="kw">let </span><font color="Maroon" title="c1">f</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">f</font> is  <a href="pencil_1.html#V15" title="PENCIL_1:attr.15">PLS-yielding</a>  implies ( <font color="Maroon" title="c1">f</font> is  <a href="waybel_3.html#V4" title="WAYBEL_3:attr.4">non-Empty</a>  &amp; <font color="Maroon" title="c1">f</font> is  <a href="pencil_1.html#V11" title="PENCIL_1:attr.11">TopStruct-yielding</a>  ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:31_1_1"/><span class="lab"><font color="Green" title="E26">A1</font></span>: 
<font color="Maroon" title="c1">f</font> is  <a href="pencil_1.html#V15" title="PENCIL_1:attr.15">PLS-yielding</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">f</font> is  <a href="waybel_3.html#V4" title="WAYBEL_3:attr.4">non-Empty</a>  &amp; <font color="Maroon" title="c1">f</font> is  <a href="pencil_1.html#V11" title="PENCIL_1:attr.11">TopStruct-yielding</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E2:31_1_1"/>
<font color="Maroon" title="c1">f</font> is  <a href="waybel_3.html#V4" title="WAYBEL_3:attr.4">non-Empty</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">f</font> is  <a href="pencil_1.html#V11" title="PENCIL_1:attr.11">TopStruct-yielding</a> </span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">S</font> be    <a href="struct_0.html#L1" title="STRUCT_0:struct.1">1-sorted</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="waybel_3.html#D7" title="WAYBEL_3:def.7">WAYBEL_3:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c2">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">proj2</a> <font color="Maroon" title="c1">f</font> or  not <font color="Maroon" title="c2">S</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:31_1_1_1"/>
<font color="Maroon" title="c2">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> not <font color="Maroon" title="c2">S</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> </span><br/>

<span class="kw">hence </span><a NAME="E2:31_1_1_1"/>
not <font color="Maroon" title="c2">S</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31_1_1"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="ref" href="pencil_1.html#D19" target="_self" title="PENCIL_1:def.19">Def19</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">let </span><font color="Maroon" title="c2">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="pencil_1.html#D13" target="_self" title="PENCIL_1:def.13">PENCIL_1:def 13</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK4" title="RELAT_1:NK.4">rng</a> <font color="Maroon" title="c1">f</font> implies <font color="Maroon" title="c2">x</font> is    <a href="pre_topc.html#L1" title="PRE_TOPC:struct.1">TopStruct</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E3:31_1_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">x</font> is    <a href="pre_topc.html#L1" title="PRE_TOPC:struct.1">TopStruct</a> </span><br/>

<span class="kw">hence </span><a NAME="E4:31_1_1"/>
<font color="Maroon" title="c2">x</font> is    <a href="pre_topc.html#L1" title="PRE_TOPC:struct.1">TopStruct</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31_1_1"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="ref" href="pencil_1.html#D19" target="_self" title="PENCIL_1:def.19">Def19</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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