reserve D,D9 for non empty set;
reserve R for Ring;
reserve G,H,S for non empty ModuleStr over R;
reserve UN for Universe;
reserve R for Ring;
reserve G, H for LeftMod of R;
reserve G1, G2, G3 for LeftMod of R;
reserve f for LModMorphismStr over R;
reserve a,b,c for Element of {0,1,2};

theorem Th24:
  for x,y,z being Scalar of Z_3, X,Y,Z being Element of {0,1,2} st
X=x & Y=y & Z=z holds (x+y)+z = (X+Y)+Z & x+(y+z) = X+(Y+Z) & (x*y)*z = (X*Y)*Z
  & x*(y*z) = X*(Y*Z)
proof
  let x,y,z be Scalar of Z_3, X,Y,Z be Element of {0,1,2};
  assume that
A1: X=x and
A2: Y=y and
A3: Z=z;
A4: x*y = X*Y & y*z = Y*Z by A1,A2,A3,Th23;
  x+y = X+Y & y+z = Y+Z by A1,A2,A3,Th23;
  hence thesis by A1,A3,A4,Th23;
end;
