reserve n for Nat;

theorem Th13:
  for n be Nat, a be Real, x1,y1 be Point of REAL-NS n,
x2,y2 be Point of REAL-US n st x1 = x2 & y1 = y2 holds x1 + y1 = x2 + y2 & -x1
  = -x2 & a * x1 = a * x2
proof
  let n be Nat, a be Real;
  let x1,y1 be Point of REAL-NS n;
  reconsider x=x1, y=y1 as Element of REAL n by Def4;
  let x2,y2 be Point of REAL-US n;
  assume that
A1: x1 = x2 and
A2: y1 = y2;
  thus x1 + y1 = (Euclid_add n).(x,y) by Def4
    .= x2 + y2 by A1,A2,Def6;
  thus -x1 = (-1)*x1 by RLVECT_1:16
    .= (Euclid_mult n).(-1,x) by Def4
    .= (-1)*x2 by A1,Def6
    .= -x2 by RLVECT_1:16;
  thus a * x1 = (Euclid_mult n).(a,x) by Def4
    .= a*x2 by A1,Def6;
end;
