reserve E, F, G,S,T,W,Y for RealNormSpace;
reserve f,f1,f2 for PartFunc of S,T;
reserve Z for Subset of S;
reserve i,n for Nat;

theorem Th5:
  for s,t be Point of [:E,F:], a be Real
  holds
      s = [s`1,s`2]
    & (s + t)`1 = s`1 + t`1 & (s + t)`2 = s`2 + t`2
    & (s - t)`1 = s`1 - t`1 & (s - t)`2 = s`2 - t`2
    & (a * s)`1 = a * s`1 & (a * s)`2 = a * s`2
  proof
    let s,t be Point of [:E,F:], a be Real;

    consider s1 be Point of E,s2 be Point of F such that
    A1: s = [s1,s2] by PRVECT_3:18;

    consider t1 be Point of E,t2 be Point of F such that
    A2: t = [t1,t2] by PRVECT_3:18;

    thus s = [s`1,s`2] by A1;
    s + t = [s`1 + t`1, s`2 + t`2] by A1,A2,PRVECT_3:18;
    hence (s + t)`1 = s`1 + t`1 & (s + t)`2 = s`2 + t`2;

    A3: (-1) * t
     = [(-1) * t`1, (-1) * t`2] by A2,PRVECT_3:18
    .= [-(t`1), (-1) * t`2] by RLVECT_1:16
    .= [-(t`1), -(t`2)] by RLVECT_1:16;

    s - t
     = s + (-1) * t by RLVECT_1:16
    .= [s`1 - t`1, s`2 - t`2] by A1,A3,PRVECT_3:18;
    hence (s - t)`1 = s`1 - t`1 & (s - t)`2 = s`2 - t`2;

    a * s = [a * s`1, a * s`2] by A1,PRVECT_3:18;
    hence (a * s)`1 = a * s`1 & (a * s)`2 = a * s`2;
  end;
