reserve Y,Z for non empty set;
reserve PA,PB for a_partition of Y;
reserve A,B for Subset of Y;
reserve i,j,k for Nat;
reserve x,y,z,x1,x2,y1,z0,X,V,a,b,d,t,SFX,SFY for set;

theorem
  for PA,PB,PC being a_partition of Y holds
  (PA '\/' PB) '\/' PC = PA '\/' (PB '\/' PC)
proof
  let PA,PB,PC be a_partition of Y;
 ERl((PA '\/' PB) '\/' PC) = ERl(PA '\/' PB) "\/" ERl(PC) by Th23
    .= (ERl(PA) "\/" ERl(PB)) "\/" ERl(PC) by Th23
    .= ERl(PA) "\/" (ERl(PB) "\/" ERl(PC)) by EQREL_1:13
    .= ERl(PA) "\/" ERl(PB '\/' PC) by Th23
    .= ERl(PA '\/' (PB '\/' PC)) by Th23;
  hence thesis by Th25;
end;
