let L be non empty satisfying_DN_1 ComplLattStr ; :: 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