let L be non empty satisfying_DN_1 ComplLLattStr ; :: thesis: for x, y, z being Element of L holds (((((x + y) `) + ((y + z) `)) `) + z) ` = (y + z) `
let x, y, z be Element of L; :: thesis: (((((x + y) `) + ((y + z) `)) `) + z) ` = (y + z) `
set Y = (((x + y) `) + ((y + z) `)) ` ;
(z + ((((x + y) `) + ((y + z) `)) `)) ` = (((((x + y) `) + ((y + z) `)) `) + z) ` by Th14;
hence (((((x + y) `) + ((y + z) `)) `) + z) ` = (y + z) ` by Th12; :: thesis: verum