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<b>let </b><font color="Maroon" title="c1">S</font> be  non <a href="struct_0.html#V11" title="STRUCT_0:attr.11">void</a> <a href="catalg_1.html#NM1" title="CATALG_1:NM.1">Signature</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>, <font color="Olive" title="b2">g</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> holds <br/> ( <font color="Olive" title="b1">f</font>,<font color="Olive" title="b2">g</font> <a href="algspec1.html#R1" title="ALGSPEC1:pred.1">form_a_replacement_in</a> <font color="Maroon" title="c1">S</font> iff the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> <a href="algspec1.html#K1" title="ALGSPEC1:func.1">-indexing</a> <font color="Olive" title="b1">f</font>,the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">S</font> <a href="algspec1.html#K1" title="ALGSPEC1:func.1">-indexing</a> <font color="Olive" title="b2">g</font> <a href="pua2mss1.html#R3" title="PUA2MSS1:pred.3">form_morphism_between</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> <font color="Olive" title="b1">f</font>,<font color="Olive" title="b2">g</font> )</span><br/><b>let </b><font color="Maroon" title="c2">f</font>, <font color="Maroon" title="c3">g</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"> ( <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font> <a href="algspec1.html#R1" title="ALGSPEC1:pred.1">form_a_replacement_in</a> <font color="Maroon" title="c1">S</font> iff the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> <a href="algspec1.html#K1" title="ALGSPEC1:func.1">-indexing</a> <font color="Maroon" title="c2">f</font>,the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">S</font> <a href="algspec1.html#K1" title="ALGSPEC1:func.1">-indexing</a> <font color="Maroon" title="c3">g</font> <a href="pua2mss1.html#R3" title="PUA2MSS1:pred.3">form_morphism_between</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font> )</span><br/>





<b>set </b><font color="Maroon" title="c4">ff</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> <a href="algspec1.html#K1" title="ALGSPEC1:func.1">-indexing</a> <font color="Maroon" title="c2">f</font>;<br/>
<b>set </b><font color="Maroon" title="c5">gg</font> = the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">S</font> <a href="algspec1.html#K1" title="ALGSPEC1:func.1">-indexing</a> <font color="Maroon" title="c3">g</font>;<br/>
<b>set </b><font color="Maroon" title="c6">S'</font> = <font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font>;<br/>
<b>thus </b><a NAME="E1:43"/>
( <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font> <a href="algspec1.html#R1" title="ALGSPEC1:pred.1">form_a_replacement_in</a> <font color="Maroon" title="c1">S</font> implies the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> <a href="algspec1.html#K1" title="ALGSPEC1:func.1">-indexing</a> <font color="Maroon" title="c2">f</font>,the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">S</font> <a href="algspec1.html#K1" title="ALGSPEC1:func.1">-indexing</a> <font color="Maroon" title="c3">g</font> <a href="pua2mss1.html#R3" title="PUA2MSS1:pred.3">form_morphism_between</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font> )
 <b>by </b><i><a class="ref" href="algspec1.html#D4" target="_self" title="ALGSPEC1:def.4">Def4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> <a href="algspec1.html#K1" title="ALGSPEC1:func.1">-indexing</a> <font color="Maroon" title="c2">f</font>,the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">S</font> <a href="algspec1.html#K1" title="ALGSPEC1:func.1">-indexing</a> <font color="Maroon" title="c3">g</font> <a href="pua2mss1.html#R3" title="PUA2MSS1:pred.3">form_morphism_between</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font> implies <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font> <a href="algspec1.html#R1" title="ALGSPEC1:pred.1">form_a_replacement_in</a> <font color="Maroon" title="c1">S</font> )</span><br/>

<b>assume </b><a NAME="E2:43"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> <a href="algspec1.html#K1" title="ALGSPEC1:func.1">-indexing</a> <font color="Maroon" title="c2">f</font>,the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">S</font> <a href="algspec1.html#K1" title="ALGSPEC1:func.1">-indexing</a> <font color="Maroon" title="c3">g</font> <a href="pua2mss1.html#R3" title="PUA2MSS1:pred.3">form_morphism_between</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</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>,<font color="Maroon" title="c3">g</font> <a href="algspec1.html#R1" title="ALGSPEC1:pred.1">form_a_replacement_in</a> <font color="Maroon" title="c1">S</font></span><br/>

<a NAME="E3:43"/><b>then </b>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> <a href="algspec1.html#K1" title="ALGSPEC1:func.1">-indexing</a> <font color="Maroon" title="c2">f</font>,the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">S</font> <a href="algspec1.html#K1" title="ALGSPEC1:func.1">-indexing</a> <font color="Maroon" title="c3">g</font> <a href="algspec1.html#R1" title="ALGSPEC1:pred.1">form_a_replacement_in</a> <font color="Maroon" title="c1">S</font>
 
<b>by </b><i><a class="ref" href="algspec1.html#T32" target="_self" title="ALGSPEC1:th.32">Th32</a></i>;<br/>
<b>hence </b><a NAME="E4:43"/>
<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font> <a href="algspec1.html#R1" title="ALGSPEC1:pred.1">form_a_replacement_in</a> <font color="Maroon" title="c1">S</font>
 <b>by </b><i><a class="ref" href="algspec1.html#T31" target="_self" title="ALGSPEC1:th.31">Th31</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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