
theorem Th3:
  for x, y being Point of linfty_Space,
   a be Real holds ( ||.x.|| =
  0 iff x = 0.linfty_Space ) & 0 <= ||.x.|| & ||.x+y.|| <= ||.x.|| + ||.y.|| &
  ||. a*x .|| = |.a.| * ||.x.||
proof
  let x, y be Point of linfty_Space;
  let a be Real;
A1: for n be Nat holds (abs(a(#)seq_id x)).n =|.a.|*(abs(
  seq_id x).n)
  proof
    let n be Nat;
    (|.a(#)seq_id x.|).n =|.(a(#)seq_id x).n.| by SEQ_1:12
      .=|.a*((seq_id x).n).| by SEQ_1:9
      .=|.a.|*(|.(seq_id x).n.|) by COMPLEX1:65
      .=|.a.|*(abs(seq_id x).n) by SEQ_1:12;
    hence thesis;
  end;
  abs(seq_id x).1 =|.(seq_id x).1.| by SEQ_1:12;
  then
A2: 0<= abs(seq_id x).1 by COMPLEX1:46;
A3: for n be Nat holds (abs seq_id(x+y)).n = |.((seq_id x).n)
  + ((seq_id y).n).|
  proof
    let n be Nat;
    (abs seq_id(x+y)).n = (abs(seq_id(seq_id x+seq_id y))).n by Th2
      .= |.(seq_id x+seq_id y).n.| by SEQ_1:12
      .= |.((seq_id x).n)+((seq_id y).n).| by SEQ_1:7;
    hence thesis;
  end;
A4: for n be Nat holds (abs seq_id(x+y)).n <= (abs seq_id x).n +
  (abs seq_id y).n
  proof
    let n be Nat;
    |.(seq_id x).n + ((seq_id y).n).| <= |.(seq_id x).n.| + |.(
    seq_id y).n.| by COMPLEX1:56;
    then
    (abs(seq_id(x+y))) .n <= |.(seq_id x).n.| + |.(seq_id y).n.| by A3;
    then (abs seq_id(x+y)) .n <= (abs seq_id x).n + |.(seq_id y).n.| by
SEQ_1:12;
    hence thesis by SEQ_1:12;
  end;
A5: for n being Nat holds (abs(seq_id(x+y))).n <= ((abs seq_id x
  ) + (abs seq_id y)).n
  proof
    let n be Nat;
    (abs seq_id x).n + (abs seq_id y).n =((abs seq_id x) + (abs seq_id y)
    ).n by SEQ_1:7;
    hence thesis by A4;
  end;
A6: now
    assume
A7: x=0.linfty_Space;
A8: for n be Nat holds (seq_id x).n=0 by A7,Th2;
    thus ||.x.|| = upper_bound rng abs seq_id x by Th2
      .= 0 by A8,Lm5;
  end;
  seq_id x is bounded by Def1;
  then
A9: 0 <= upper_bound rng abs seq_id x by A2,Lm2;
  seq_id x is bounded by Def1;
  then rng abs seq_id x is real-bounded by MEASURE6:39;
  then
A10: rng abs seq_id x is bounded_above;
A11: now
A12: x in the_set_of_RealSequences by Def1;
    assume
A13: ||.x.|| = 0;
    ||.x.|| = upper_bound rng abs seq_id x & seq_id x is bounded by Th2;
    then for n be Nat holds 0 = (seq_id x).n by A13,Lm6;
    hence x=0.linfty_Space by A12,Th2,RSSPACE:5;
  end;
A14: seq_id y is bounded by Def1;
A15: seq_id x is bounded by Def1;
  now
    let n be Nat;
A16: (abs seq_id y).n <=upper_bound rng abs seq_id y by A14,Lm2;
    (abs seq_id x + abs seq_id y).n = (abs seq_id x).n + (abs seq_id y).n
    & (abs seq_id x).n <=upper_bound rng abs seq_id x by A15,Lm2,SEQ_1:7;
    then
A17: ((abs seq_id x) + (abs seq_id y)).n <= upper_bound(rng abs seq_id x)
    + upper_bound rng abs seq_id y by A16,XREAL_1:7;
    (abs seq_id(x+y)).n <= (abs seq_id x + abs seq_id y).n by A5;
    hence
    (abs seq_id(x+y)).n <= upper_bound rng abs seq_id x + upper_bound rng
    abs seq_id y by A17,XXREAL_0:2;
  end;
  then
A18: upper_bound rng abs seq_id(x+y) <= upper_bound rng abs seq_id x +
  upper_bound rng abs seq_id y by Lm1;
A19: ||.y.|| = upper_bound rng abs seq_id y &
upper_bound rng abs seq_id(x+y) = ||.x + y.||
  by Th2;
  ||. a*x .|| = upper_bound rng abs seq_id(a*x) by Th2
    .=upper_bound rng |.seq_id(a(#)seq_id x).| by Th2
    .=upper_bound(rng (|.a.| (#) (abs seq_id x))) by A1,SEQ_1:9
    .=upper_bound(|.a.|**rng abs seq_id x) by INTEGRA2:17
    .=|.a.|*upper_bound rng abs seq_id x by A10,COMPLEX1:46,INTEGRA2:13
    .=|.a.|*||.x.|| by Th2;
  hence thesis by A11,A6,A9,A19,A18,Th2;
end;
