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 Th22:
  a*(b1*x1+b2*x2+b3*x3) = (a*b1)*x1+(a*b2)*x2+(a*b3)*x3
proof
  thus a*(b1*x1+b2*x2+b3*x3) = a*(b1*x1)+a*(b2*x2)+a*(b3*x3) by Th20
    .= (a*b1)*x1+a*(b2*x2)+a*(b3*x3) by EUCLID_4:4
    .= (a*b1)*x1+(a*b2)*x2+a*(b3*x3) by EUCLID_4:4
    .= (a*b1)*x1+(a*b2)*x2+(a*b3)*x3 by EUCLID_4:4;
end;
