let L be non empty satisfying_DN_1 ComplLattStr ; :: thesis: for x, y, z being Element of L holds (((x + y) ` ) + ((((z + x) ` ) + y) ` )) ` = y
let x, y, z be Element of L; :: thesis: (((x + y) ` ) + ((((z + x) ` ) + y) ` )) ` = y
consider u being Element of L;
set U = ((y ` ) + ((y + u) ` )) ` ;
(((x + y) ` ) + ((((z + x) ` ) + ((((((y + (y ` )) ` ) + y) ` ) + ((y + (((y ` ) + ((y + u) ` )) ` )) ` )) ` )) ` )) ` = y by Th4;
hence (((x + y) ` ) + ((((z + x) ` ) + y) ` )) ` = y by Def1; :: thesis: verum