let p, q, r be Element of LTLB_WFF ; :: thesis: (p => r) => ((q => r) => ((p 'or' q) => r)) is ctaut
let g be Function of LTLB_WFF,BOOLEAN; :: according to LTLAXIO1:def 16 :: thesis: (VAL g) . ((p => r) => ((q => r) => ((p 'or' q) => r))) = 1
set v = VAL g;
A1: ( (VAL g) . p = 1 or (VAL g) . p = 0 ) by XBOOLEAN:def 3;
A2: ( (VAL g) . r = 1 or (VAL g) . r = 0 ) by XBOOLEAN:def 3;
A3: (VAL g) . (p => r) = ((VAL g) . p) => ((VAL g) . r) by LTLAXIO1:def 15;
A4: ( (VAL g) . q = 1 or (VAL g) . q = 0 ) by XBOOLEAN:def 3;
(VAL g) . ((q => r) => ((p 'or' q) => r)) = ((VAL g) . (q => r)) => ((VAL g) . ((p 'or' q) => r)) by LTLAXIO1:def 15
.= (((VAL g) . q) => ((VAL g) . r)) => ((VAL g) . ((p 'or' q) => r)) by LTLAXIO1:def 15
.= (((VAL g) . q) => ((VAL g) . r)) => (((VAL g) . (p 'or' q)) => ((VAL g) . r)) by LTLAXIO1:def 15
.= (((VAL g) . q) => ((VAL g) . r)) => ((((VAL g) . p) 'or' ((VAL g) . q)) => ((VAL g) . r)) by Th5 ;
hence (VAL g) . ((p => r) => ((q => r) => ((p 'or' q) => r))) = (((VAL g) . p) => ((VAL g) . r)) => ((((VAL g) . q) => ((VAL g) . r)) => ((((VAL g) . p) 'or' ((VAL g) . q)) => ((VAL g) . r))) by LTLAXIO1:def 15, A3
.= 1 by A1, A2, A4 ;
:: thesis: verum