
theorem
  for G be RealNormSpace-Sequence,
    p,q,r be Point of product G
  holds
    p-q = r
  iff
    for i be Element of dom G holds r.i = p.i - q.i
  proof
    let G be RealNormSpace-Sequence,
      p,q,r be Point of product G;
    reconsider p0 = p, q0 = q, r0 = r as Element of product carr G by EXTh10;
    reconsider qq0 = (-1)*q as Element of product carr G by EXTh10;
    A2: p-q = p+(-1)*q by RLVECT_1:16;
    hereby
      assume
      A3: p-q = r;
      thus for i be Element of dom G holds r.i = p.i - q.i
      proof
        let i be Element of dom G;
        A4: r0.i = p0.i + qq0.i by EXTh12,A3,A2;
        qq0.i = (-1)*(q0.i) by EXTh13;
        hence thesis by A4,RLVECT_1:16;
      end;
    end;
    assume
    A5: for i be Element of dom G holds r.i = p.i - q.i;
    now
      let i be Element of dom G;
      A6: qq0.i = (-1)*(q0.i) by EXTh13;
      r0.i = p0.i - q0.i by A5;
      hence r0.i = p0.i + qq0.i by A6,RLVECT_1:16;
    end;
    hence p-q = r by A2,EXTh12;
  end;
