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 Th26:
  (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 Th18
    .= ((a1-b1)*x1+(a2-b2)*x2)+(a3*x3-b3*x3) by Th25
    .= (a1-b1)*x1+(a2-b2)*x2+(a3-b3)*x3 by Th11;
end;
