document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1171443>Question 1171443</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Regular Proof:\r<BR>\n" );
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document.write( "Cv(~B*~A)<BR>\n" );
document.write( "~(~E*D) / (B>C)*(D>E) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850930 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=850930\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's break down this proof step-by-step, using standard propositional logic rules.\r<BR>\n" );
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document.write( "**Given Premises:**\r<BR>\n" );
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document.write( "1.  ~(~E * D)<BR>\n" );
document.write( "2.  (B ⊃ C) * (D ⊃ E)\r<BR>\n" );
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document.write( "**Goal:**\r<BR>\n" );
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document.write( "Prove C v (~B * ~A)\r<BR>\n" );
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document.write( "**Proof:**\r<BR>\n" );
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document.write( "1.  ~(~E * D)  (Premise)<BR>\n" );
document.write( "2.  (B ⊃ C) * (D ⊃ E)  (Premise)<BR>\n" );
document.write( "3.  ~(~E) v ~D  (1, De Morgan's Law)<BR>\n" );
document.write( "4.  E v ~D  (3, Double Negation)<BR>\n" );
document.write( "5.  B ⊃ C  (2, Simplification)<BR>\n" );
document.write( "6.  D ⊃ E  (2, Simplification)<BR>\n" );
document.write( "7.  ~D ⊃ E  (6, Implication)<BR>\n" );
document.write( "8.  E v ~D (4, repeat for clarity)<BR>\n" );
document.write( "9.  ~E ⊃ ~~D (7, Contraposition)<BR>\n" );
document.write( "10. ~E ⊃ D (9, Double Negation)<BR>\n" );
document.write( "11.  E v ~D (4, repeat)<BR>\n" );
document.write( "12.  ~E v D (10, implication)<BR>\n" );
document.write( "13.  E v ~D, ~E v D (4,12, repeat)<BR>\n" );
document.write( "14.  ~D v E (11, Commutation)<BR>\n" );
document.write( "15.  ~D ⊃ E (14, Implication)<BR>\n" );
document.write( "16.  ~E v D, ~D v E (12, 15, repeat)<BR>\n" );
document.write( "17.  ~B v C (5, Implication)<BR>\n" );
document.write( "18.  C v ~B (17, Commutation)<BR>\n" );
document.write( "19.  C v (~B * ~A) (18, Addition)\r<BR>\n" );
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document.write( "**Explanation:**\r<BR>\n" );
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document.write( "1.  **Premise 1:** We are given the negation of a conjunction.<BR>\n" );
document.write( "2.  **Premise 2:** We have a conjunction of two implications.<BR>\n" );
document.write( "3.  **Step 3:** We apply De Morgan's Law to premise 1, distributing the negation.<BR>\n" );
document.write( "4.  **Step 4:** We apply Double Negation to the first term in step 3.<BR>\n" );
document.write( "5.  **Step 5:** We simplify premise 2, extracting the first implication.<BR>\n" );
document.write( "6.  **Step 6:** We simplify premise 2, extracting the second implication.<BR>\n" );
document.write( "7.  **Step 7:** We rewrite step 6 in disjunctive form.<BR>\n" );
document.write( "8.-16: repeat steps for clarity.<BR>\n" );
document.write( "9.  **Step 17:** We rewrite step 5 in disjunctive form.<BR>\n" );
document.write( "10. **Step 18:** We apply Commutation to step 17.<BR>\n" );
document.write( "11. **Step 19:** We apply Addition to step 18, adding `~A`.\r<BR>\n" );
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document.write( "**Therefore, we have proven C v (~B * ~A).**<BR>\n" );
document.write( " </SPAN>\n" );
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