
theorem Th7:
  for x, y being Point of l1_Space, a be Real holds ( ||.x.|| = 0
iff x = 0.l1_Space ) & 0 <= ||.x.|| & ||.x+y.|| <= ||.x.|| + ||.y.|| & ||. a*x
  .|| = |.a.| * ||.x.||
proof
  let x, y be Point of l1_Space;
  let a be Real;
A1: for n be Nat holds 0 <= abs(seq_id x).n
  proof
    let n be Nat;
    0 <= |.(seq_id x).n.| by COMPLEX1:46;
    hence thesis by SEQ_1:12;
  end;
A2: now
    let n be Nat;
    (abs(seq_id(x+y))).n = |.(seq_id(x+y)).n.| by SEQ_1:12;
    hence 0 <= (abs(seq_id(x+y))).n by COMPLEX1:46;
  end;
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 Th6
      .= |.(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;
  seq_id y is absolutely_summable by Def1;
  then
A6: abs seq_id y is summable by SERIES_1:def 4;
  seq_id x is absolutely_summable by Def1;
  then abs seq_id x is summable by SERIES_1:def 4;
  then (abs seq_id x) + (abs seq_id y) is summable by A6,SERIES_1:7;
  then
A7: Sum(abs(seq_id(x+y))) <= Sum((abs seq_id x) + (abs seq_id y)) by A5,A2,
SERIES_1:20;
A8: now
    assume x=0.l1_Space;
    then
A9: for n be Nat holds (seq_id x).n=0 by Th6;
    thus ||.x.|| = Sum abs seq_id x by Def2
      .= 0 by A9,Th3;
  end;
A10: Sum abs seq_id(x+y) = ||.x + y.|| by Th6;
A11: now
A12: x in the_set_of_RealSequences by Def1;
    assume
A13: ||.x.|| = 0;
    ||.x.|| = Sum abs seq_id x & seq_id(x) is absolutely_summable by Th6;
    then for n be Nat holds 0 = (seq_id x).n by A13,Th4;
    hence x=0.l1_Space by A12,Th6,RSSPACE:5;
  end;
A14: ||.x.|| = Sum abs seq_id x & ||.y.|| = Sum abs seq_id y by Th6;
A15: for n be Nat holds (abs(a(#)seq_id(x))).n =|.a.|*(abs(
  seq_id x).n)
  proof
    let n be Nat;
    (abs(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;
  seq_id x is absolutely_summable by Def1;
  then
A16: abs seq_id x is summable by SERIES_1:def 4;
  seq_id x is absolutely_summable by Def1;
  then
A17: abs seq_id x is summable by SERIES_1:def 4;
  ||.(a*x).|| =Sum(abs(seq_id(a*x))) by Th6
    .=Sum(|.seq_id(a(#)seq_id(x)).|) by Th6
    .=Sum(|.a.| (#) (abs seq_id x)) by A15,SEQ_1:9
    .=|.a.|*Sum(abs seq_id x) by A16,SERIES_1:10
    .=|.a.|*||.x.|| by Th6;
  hence thesis by A11,A8,A1,A17,A14,A10,A6,A7,SERIES_1:7,18;
end;
