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 Th23:
  for x,y being Scalar of Z_3, X,Y being Element of {0,1,2} st X=x
  & Y=y holds x+y = X+Y & x*y = X*Y & -x = -X
proof
  let x,y be Scalar of Z_3, X,Y be Element of {0,1,2};
  assume that
A1: X=x and
A2: Y=y;
  thus x+y = X+Y by A1,A2,Def15;
  thus x*y = X*Y by A1,A2,Def16;
  reconsider a = -X as Element of Z_3;
  x + a = X + -X by A1,Def15
    .= 0.Z_3 by Lm7;
  hence thesis by RLVECT_1:def 10;
end;
