By definition of $\impliedby$, we prove: :$\vdash \left({\left({p \lor q}\right) \lor r}\right) \implies \left({p \lor \left({q \lor r}\right)}\right)$ {{BeginTableau|\vdash \left({\left({p \lor q}\right) \lor r}\right) \implies \left({p \lor \left({q \lor r}\right)}\right)|Instance 2 of the Hilbert-style systems}} {{TableauLine  |n = 1  |f = r \implies \paren{ q \lor r }  |rlnk = Definition:Hilbert Proof System/Instance 2  |rtxt = Axiom $A2$  |c = $r/q, q/p$ }} {{TableauLine  |n = 2  |f = \paren{ r \implies \paren{ q \lor r } } \implies \paren{ \paren{ \paren{ p \lor q } \lor r } \implies \paren{ \paren{ p \lor q } \lor \paren{ q \lor r } } }  |rlnk = Definition:Hilbert Proof System/Instance 2  |rtxt = Axiom $A4$  |c = $r/q, \paren{ p \lor q } \,/\, p, \paren{ q \lor r } \,/\, r$ }} {{TableauLine  |n = 3  |f = \paren{ \paren{ p \lor q } \lor r } \implies \paren{ \paren{ p \lor q } \lor \paren{ q \lor r } }  |rlnk = Definition:Hilbert Proof System/Instance 2  |rtxt = Rule $RST \, 3$  |dep  = 1,2 }} <!-- D5 --> {{TableauLine  |n = 4  |f = q \implies \paren{ r \lor q }  |rlnk = Definition:Hilbert Proof System/Instance 2  |rtxt = Axiom $A2$  |c = $r/p$ }} {{TableauLine  |n = 5  |f = \paren{ q \implies \paren{ r \lor q } } \implies \paren{ \paren{ p \lor q } \implies \paren{ p \lor \paren{ r \lor q } } }  |rlnk = Definition:Hilbert Proof System/Instance 2  |rtxt = Axiom $A4$  |c = $\paren{ r \lor q } \,/\, r$ }} {{TableauLine  |n = 6  |f = \paren{ p \lor q } \implies \paren{ p \lor \paren{ r \lor q } }  |rlnk = Definition:Hilbert Proof System/Instance 2  |rtxt = Rule $RST \, 3$  |dep  = 4,5 }} {{TableauLine  |n = 7  |f = \paren{ p \lor \paren{ r \lor q } } \implies \paren{ \paren{ r \lor q } \lor p }  |rlnk = Definition:Hilbert Proof System/Instance 2  |rtxt = Axiom $A3$  |c = $\paren{ r \lor q } \,/\, q$ }} {{TableauLine  |n = 8  |f = \paren{ p \lor q } \implies \paren{ \paren{ r \lor q } \lor p }  |rlnk = Hypothetical Syllogism/Formulation 1/Proof 3  |rtxt = Hypothetical Syllogism  |dep  = 6,7 }} <!-- D6 --> {{TableauLine  |n = 9  |f = q \implies \paren{ q \lor p }  |rlnk = Rule of Addition/Sequent Form/Formulation 2/Form 1/Proof 2  |rtxt = Rule of Addition  |c = $p \,/\, q, q \,/\, p$ }} {{TableauLine  |n = 10  |f = q \implies \paren{ q \lor s }  |rlnk = Rule of Addition/Sequent Form/Formulation 2/Form 1/Proof 2  |rtxt = Rule of Addition  |c = $s \,/\, q, q \,/\, p$ }} {{TableauLine  |n = 11  |f = \paren{ q \implies \paren{ q \lor s } } \implies \paren{ \paren{ p \lor q } \implies \paren{ p \lor \paren{ q \lor s } } }  |rlnk = Definition:Hilbert Proof System/Instance 2  |rtxt = Axiom $A4$  |c = $\paren{ q \lor s } \,/\, r$ }} {{TableauLine  |n = 12  |f = \paren{ p \lor q } \implies \paren{ p \lor \paren{ q \lor s } }  |rlnk = Definition:Hilbert Proof System/Instance 2  |rtxt = Rule $RST \, 3$  |dep  = 10,11 }} {{TableauLine  |n = 13  |f = \paren{ \paren{ p \lor q } \implies \paren{ p \lor \paren{ q \lor s } } } \implies \paren{ \paren{ r \lor \paren{ p \lor q } } \implies \paren{ r \lor \paren{ p \lor \paren{ q \lor s } } } }  |rlnk = Definition:Hilbert Proof System/Instance 2  |rtxt = Axiom $A4$  |c = $r \,/\, p, \paren{ p \lor q } \,/\, q, \paren{ p \lor paren{ q \lor s } } \,/\, r$ }} {{TableauLine  |n = 14  |f = \paren{ r \lor \paren{ p \lor q } } \implies \paren{ r \lor \paren{ p \lor \paren{ q \lor s } } }  |rlnk = Definition:Hilbert Proof System/Instance 2  |rtxt = Rule $RST \, 3$  |dep  = 12,13 }} <!-- D7 --> {{TableauLine  |n = 15  |f = \paren{ \paren{ p \lor q } \lor \paren{ q \lor r } } \implies \paren{ \paren{ p \lor \paren{ q \lor r } } \lor \paren{ p \lor q } }  |rlnk = Definition:Hilbert Proof System/Instance 2  |rtxt = Rule $RST \, 1$  |dep = 8  |c = $\paren{ p \lor q } \,/\, p, \paren{ q \lor r } \,/\, q, p \,/\, r$ }} {{TableauLine  |n = 16  |f = \paren{ \paren{ p \lor \paren{ q \lor r } } \lor \paren{ p \lor q } } \implies \paren{ \paren{ p \lor \paren{ q \lor r } } \lor \paren{ p \lor \paren{ q \lor r } } }  |rlnk = Definition:Hilbert Proof System/Instance 2  |rtxt = Rule $RST \, 1$  |dep = 14  |c = $\paren{ p \lor \paren{ q \lor r } \,/\, r, r \,/\, s$ }} {{TableauLine  |n = 17  |f = \paren{ \paren{ p \lor \paren{ q \lor r } } \lor \paren{ p \lor \paren{ q \lor r } } } \implies \paren{ p \lor \paren{ q \lor r } }  |rlnk = Definition:Hilbert Proof System/Instance 2  |rtxt = Axiom $A1$  |c = $\paren{ p \lor \paren{ q \lor r } } \,/\, p$ }} {{TableauLine  |n = 18  |f = \paren{ \paren{ p \lor q } \lor \paren{ q \lor r } } \implies \paren{ \paren{ p \lor \paren{ q \lor r } } \lor \paren{ p \lor \paren{ q \lor r } } }  |rlnk = Hypothetical Syllogism/Formulation 1/Proof 3  |rtxt = Hypothetical Syllogism  |dep  = 15,16 }} {{TableauLine  |n = 19  |f = \paren{ \paren{ p \lor q } \lor \paren{ q \lor r } } \implies \paren{ p \lor \paren{ q \lor r } }  |rlnk = Hypothetical Syllogism/Formulation 1/Proof 3  |rtxt = Hypothetical Syllogism  |dep  = 17,18 }}<!-- D8 --> {{TableauLine  |n = 20  |f = \paren{ \paren{ p \lor q } \lor r } \implies \paren{ p \lor \paren{ q \lor r } }  |rlnk = Hypothetical Syllogism/Formulation 1/Proof 3  |rtxt = Hypothetical Syllogism  |dep  = 3,19 }}<!-- D9 --> {{EndTableau}} {{qed}}