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
  x - (s+t+u)*y = x + (-s)*y + (-t)*y + (-u)*y
proof
  thus x - (s+t+u)*y = x - ((s+t)*y + u*y) by EUCLID_4:7
    .= x - (s*y + t*y + u*y) by EUCLID_4:7
    .= x + ((-s)*y + (-t)*y + (-u)*y) by Th14
    .= x + ((-s)*y + (-t)*y) + (-u)*y by RVSUM_1:15
    .= x + (-s)*y + (-t)*y + (-u)*y by RVSUM_1:15;
end;
