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 Th12:
  a*(x - y) = a*x + -a*y & a*(x - y) = a*x + (-a)*y & a*(x - y) = a*x - a*y
proof
  thus
A1: a*(x - y) = a*x + a*(-y) by EUCLID_4:6
    .= a*x + -a*y by Th3;
  hence a*(x - y) = a*x + (-a)*y by Th3;
  thus thesis by A1;
end;
