
theorem Th6:
  for x, y being Point of Complex_l1_Space, p be Complex holds (
||.x.|| = 0 iff x = 0.Complex_l1_Space ) & 0 <= ||.x.|| & ||.x+y.|| <= ||.x.||
  + ||.y.|| & ||. p*x .|| = |.p.| * ||.x.||
proof
  let x, y be Point of Complex_l1_Space;
  let p be Complex;
A1: for n be Nat holds 0 <= (|.seq_id x.|).n
  proof
    let n be Nat;
    0 <= |.((seq_id x).n).| by COMPLEX1:46;
    hence thesis by VALUED_1:18;
  end;
A2: now
    let n be Nat;
    (|.seq_id(x+y).|).n = |.((seq_id(x+y)).n).| by VALUED_1:18;
    hence 0 <= (|.seq_id(x+y).|).n by COMPLEX1:46;
  end;
A3: for n be Nat holds (|.seq_id(x+y).|).n = |.(((seq_id x).n) +
  ((seq_id y).n)).|
  proof
    let n be Nat;
A4:  n in NAT by ORDINAL1:def 12;
    (|.seq_id(x+y).|).n = (|.(seq_id(seq_id(x)+seq_id y)).|).n by Th5
      .= |.((seq_id x+seq_id y).n).| by VALUED_1:18
      .= |.(((seq_id x).n)+((seq_id y).n)).| by VALUED_1:1,A4;
    hence thesis;
  end;
A5: for n be Nat holds (|.seq_id(x+y).|).n <= (|.seq_id x.|).n +
  (|.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 (|.(seq_id(x+y)).|) .n <= |.((seq_id x).n).| + |.((seq_id y).n).| by
A3;
    then (|.(seq_id(x+y)).|) .n <= (|.(seq_id x).|).n + |.((seq_id y).n).| by
VALUED_1:18;
    hence thesis by VALUED_1:18;
  end;
A6: for n being Nat holds (|.seq_id(x+y).|).n <= (|.seq_id x.| +
  |.seq_id y.|).n
  proof
    let n be Nat;
    (|.seq_id x.|).n + (|.seq_id y.|).n =((|.seq_id x.|) + (|.seq_id y.|)
    ).n by SEQ_1:7;
    hence thesis by A5;
  end;
  seq_id y is absolutely_summable by Def1;
  then
A7: |.seq_id y.| is summable by COMSEQ_3:def 9;
  seq_id x is absolutely_summable by Def1;
  then |.seq_id x.| is summable by COMSEQ_3:def 9;
  then |.seq_id x.| + |.seq_id y.| is summable by A7,SERIES_1:7;
  then
A8: Sum(|.seq_id(x+y).|) <= Sum(|.seq_id x.| + |.seq_id y.|) by A6,A2,
SERIES_1:20;
A9: now
    assume x=0.Complex_l1_Space;
    then
A10: for n be Nat holds (seq_id x).n=0 by Th5,CSSPACE:4;
    thus ||.x.|| = Sum |.seq_id x.| by Def2
      .= 0 by A10,Th3;
  end;
A11: Sum |.seq_id(x+y).| = ||.x + y.|| by Th5;
A12: now
A13: x in the_set_of_ComplexSequences by Def1;
    assume
A14: ||.x.|| = 0;
    ||.x.|| = Sum |.seq_id x.| & seq_id(x) is absolutely_summable by Th5;
    then for n be Nat holds 0 = (seq_id x).n by A14,CSSPACE2:13;
    hence x=0.Complex_l1_Space by A13,Th5,CSSPACE:5;
  end;
A15: ||.x.|| = Sum |.seq_id x.| & ||.y.|| = Sum |.seq_id y.| by Th5;
A16: for n be Nat holds (|.p(#)seq_id(x).|).n =|.p.|*(|.seq_id x
  .| .n)
  proof
    let n be Nat;
    reconsider p as Element of COMPLEX by XCMPLX_0:def 2;
    (|.(p(#)seq_id(x)).|).n =|.((p(#)seq_id(x)).n).| by VALUED_1:18
      .=|.(p*((seq_id x).n)).| by VALUED_1:6
      .=(|.p.|)*(|.((seq_id x).n).|) by COMPLEX1:65
      .=(|.p.|)*((|.seq_id x.|).n) by VALUED_1:18;
    hence thesis;
  end;
  seq_id x is absolutely_summable by Def1;
  then
A17: |.seq_id x.| is summable by COMSEQ_3:def 9;
  seq_id x is absolutely_summable by Def1;
  then
A18: |.seq_id x.| is summable by COMSEQ_3:def 9;
  ||.(p*x).|| =Sum(|.seq_id(p*x).|) by Th5
    .=Sum(|.seq_id(p(#)seq_id(x)).|) by Th5
    .=Sum(|.p.|(#)|.seq_id x.|) by A16,SEQ_1:9
    .=|.p.|*Sum(|.seq_id x.|) by A17,SERIES_1:10
    .=|.p.|*||.x.|| by Th5;
  hence thesis by A12,A9,A1,A18,A15,A11,A7,A8,SERIES_1:7,18;
end;
