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