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<span class="kw">let </span><font color="Maroon" title="c1">f</font> be   <a href="pencil_1.html#V11" title="PENCIL_1:attr.11">TopStruct-yielding</a>  <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="pencil_1.html#V14" title="PENCIL_1:attr.14">non-Trivial-yielding</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:31_3_1"/><span class="lab"><font color="Green" title="E26">A3</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="pencil_1.html#V14" title="PENCIL_1:attr.14">non-Trivial-yielding</a> </span><br/>

<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#D18" target="_self" title="PENCIL_1:def.18">PENCIL_1:def 18</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#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">f</font> implies <font color="Maroon" title="c2">x</font> is  non  <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a>   <a href="struct_0.html#L1" title="STRUCT_0:struct.1">1-sorted</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E2:31_3_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  non  <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a>   <a href="struct_0.html#L1" title="STRUCT_0:struct.1">1-sorted</a> </span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c3">S1</font> = <font color="Maroon" title="c2">x</font> as  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  non  <a href="pencil_1.html#V3" title="PENCIL_1:attr.3">void</a>   <a href="pencil_1.html#V5" title="PENCIL_1:attr.5">with_non_trivial_blocks</a>   <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_3_1"><span class="lab"><font color="Green" title="E26">A3</font></span></a>, <a class="ref" href="pencil_1.html#D19" target="_self" title="PENCIL_1:def.19">Def19</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c4">s</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E5:31_3_1"/><span class="lab"><font color="Green" title="E27">A4</font></span>: 
<font color="Maroon" title="c4">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c3">S1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">s</font> = <font color="Maroon" title="c4">s</font> as   <a href="pencil_1.html#NM1" title="PENCIL_1:NM.1">Block</a> of <font color="Maroon" title="c3">S1</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:31_3_1"><span class="lab"><font color="Green" title="E27">A4</font></span></a></span>;<br/>
<a NAME="E7:31_3_1"/>
2 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c5">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pencil_1.html#D6" target="_self" title="PENCIL_1:def.6">Def6</a></span>;<br/>
<a NAME="E8:31_3_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">s1</font>, <font color="Olive" title="b2">s2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">s</font> &amp; <font color="Olive" title="b2">s2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">s</font> &amp; <font color="Olive" title="b1">s1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">s2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pencil_1.html#T2" target="_self" title="PENCIL_1:th.2">Th2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:31_3_1"/>
<font color="Maroon" title="c2">x</font> is  non  <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a>   <a href="struct_0.html#L1" title="STRUCT_0:struct.1">1-sorted</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:31_3_1"><span class="lab"><font color="Green" title="E27">A4</font></span></a>, <a class="ref" href="struct_0.html#D10" title="STRUCT_0:def.10">STRUCT_0:def 10</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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