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 Th11:
  (a - b)*x = a*x + (-b)*x & (a - b)*x = a*x + -b*x & (a - b)*x = a*x - b*x
proof
  thus
A1: (a - b)*x = (a + -b)*x .= a*x + (-b)*x by EUCLID_4:7;
  hence (a - b)*x = a*x + -b*x by Th3;
  thus thesis by A1,Th3;
end;
