let L be non empty satisfying_DN_1 ComplLLattStr ; :: thesis: for x, y, z being Element of L holds (((((((x + y) ` ) + z) ` ) + (((x ` ) + y) ` )) ` ) + y) ` = ((x ` ) + y) `
let x, y, z be Element of L; :: thesis: (((((((x + y) ` ) + z) ` ) + (((x ` ) + y) ` )) ` ) + y) ` = ((x ` ) + y) `
set X = (x + y) ` ;
set Z = ((x ` ) + y) ` ;
set Y = z;
(((((((x + y) ` ) + z) ` ) + (((x ` ) + y) ` )) ` ) + ((((x + y) ` ) + (((x ` ) + y) ` )) ` )) ` = ((x ` ) + y) ` by Th9;
hence (((((((x + y) ` ) + z) ` ) + (((x ` ) + y) ` )) ` ) + y) ` = ((x ` ) + y) ` by Th6; :: thesis: verum