reserve q,r,c,c1,c2,c3 for Quaternion;
reserve x1,x2,x3,x4,y1,y2,y3,y4 for Real;

theorem Th2:
  c1 + c2 + c3 = c1 + (c2 + c3)
proof
  consider x1,y1,w1,z1 be Element of REAL such that
A1: c1 = [*x1,y1,w1,z1*] by Lm1;
  consider x2,y2,w2,z2 be Element of REAL such that
A2: c2 = [*x2,y2,w2,z2*] by Lm1;
  consider x3,y3,w3,z3 be Element of REAL such that
A3: c3 = [*x3,y3,w3,z3*] by Lm1;
A4: c2+c3=[*x2+x3,y2+y3,w2+w3,z2+z3*] by A2,A3,QUATERNI:def 7;
  c1+c2+c3 =[*x1+x2,y1+y2,w1+w2,z1+z2*]+[*x3,y3,w3,z3*]
  by A1,A2,A3,QUATERNI:def 7
    .=[*(x1+x2)+x3,(y1+y2)+y3,(w1+w2)+w3,(z1+z2)+z3*]
  by QUATERNI:def 7
    .=[*x1+(x2+x3),y1+(y2+y3),w1+(w2+w3),z1+(z2+z3)*]
    .=c1+(c2+c3) by A1,A4,QUATERNI:def 7;
  hence thesis;
end;
