let p, q, r be Element of PL-WFF ; :: thesis: (p => (q => r)) => ((p => q) => (p => r)) is tautology
let M be PLModel; :: according to PL_AXIOM:def 18 :: thesis: (SAT M) . ((p => (q => r)) => ((p => q) => (p => r))) = 1
thus (SAT M) . ((p => (q => r)) => ((p => q) => (p => r))) = ((SAT M) . (p => (q => r))) => ((SAT M) . ((p => q) => (p => r))) by Def11
.= (((SAT M) . p) => ((SAT M) . (q => r))) => ((SAT M) . ((p => q) => (p => r))) by Def11
.= (((SAT M) . p) => (((SAT M) . q) => ((SAT M) . r))) => ((SAT M) . ((p => q) => (p => r))) by Def11
.= (((SAT M) . p) => (((SAT M) . q) => ((SAT M) . r))) => (((SAT M) . (p => q)) => ((SAT M) . (p => r))) by Def11
.= (((SAT M) . p) => (((SAT M) . q) => ((SAT M) . r))) => ((((SAT M) . p) => ((SAT M) . q)) => ((SAT M) . (p => r))) by Def11
.= (((SAT M) . p) => (((SAT M) . q) => ((SAT M) . r))) => ((((SAT M) . p) => ((SAT M) . q)) => (((SAT M) . p) => ((SAT M) . r))) by Def11
.= 1 by XBOOLEAN:109 ; :: thesis: verum