reserve a,a1,a2,a3,b,b1,b2,b3,r,s,t,u for Real;
reserve n for Nat;
reserve x0,x,x1,x2,x3,y0,y,y1,y2,y3 for Element of REAL n;

theorem Th24:
  (a1*x1+a2*x2+a3*x3)+(b1*x1+b2*x2+b3*x3)=(a1+b1)*x1+(a2+b2)*x2+( a3+b3)*x3
proof
  thus (a1*x1+a2*x2+a3*x3)+(b1*x1+b2*x2+b3*x3) = ((a1*x1+a2*x2)+(b1*x1+b2*x2))
  +(a3*x3+b3*x3) by Th16
    .= (a1+b1)*x1+(a2+b2)*x2+(a3*x3+b3*x3) by Th23
    .= (a1+b1)*x1+(a2+b2)*x2+(a3+b3)*x3 by EUCLID_4:7;
end;
