----- Path 1: -----
        Main goal to prove:
(GOAL)  A or (B or C) => (A or B) or C
=>I     To show (GOAL) it suffices to show (SG1) assuming (ASS1):
(ASS1)  A or (B or C)
(SG1)   (A or B) or C
orE     By (ASS1) we have to show (SG1) for two cases:
        Case 1: Show (SG2) assuming (ASS2). See path 2 for case 2.
(ASS2)  A
(SG2)   (A or B) or C
orI2    To show (SG2) it suffices to show (SG3) assuming (ASS3):
(ASS3)  ~C
(SG3)   A or B
orI2    To show (SG3) it suffices to show (SG4) assuming (ASS4):
(ASS4)  ~B
(SG4)   A
Kn      By (SG4) and (ASS2) the path is complete.
(SG5)   <nothing>
----- Path 2: -----
...
orE     By (ASS1) we have to show (SG1) for two cases:
        Case 2: Show (SG6) assuming (ASS5). See path 1 for case 1.
(ASS5)  B or C
(SG6)   (A or B) or C
orE     By (ASS5) we have to show (SG6) for two cases:
        Case 1: Show (SG7) assuming (ASS6). See path 3 for case 2.
(ASS6)  B
(SG7)   (A or B) or C
orI2    To show (SG7) it suffices to show (SG8) assuming (ASS7):
(ASS7)  ~C
(SG8)   A or B
orI1    To show (SG8) it suffices to show (SG9) assuming (ASS8):
(ASS8)  ~A
(SG9)   B
Kn      By (SG9) and (ASS6) the path is complete.
(SG10)  <nothing>
----- Path 3: -----
...
orE     By (ASS5) we have to show (SG6) for two cases:
        Case 2: Show (SG11) assuming (ASS9). See path 2 for case 1.
(ASS9)  C
(SG11)  (A or B) or C
orI1    To show (SG11) it suffices to show (SG12) assuming (ASS10):
(ASS10) ~(A or B)
(SG12)  C
Kn      By (SG12) and (ASS9) the path is complete.
(SG13)  <nothing>
----- End of proof-tree -----