
theorem Th2:
  for x, y, z being Point of l2_Space for a be Real holds
  ( x .|. x = 0 iff x = 0.l2_Space ) &
  0 <= x .|. x & x .|. y = y .|. x & (x+y) .|. z = x .|. z + y .|. z &
  (a*x) .|. y = a * ( x .|. y )
proof
  let x, y, z be Point of l2_Space;
  let a be Real;
A1: for n be Nat holds 0 <= (seq_id(x)(#)seq_id(x)).n
  proof
    let n be Nat;
    (seq_id(x)(#)seq_id(x)).n =(seq_id x).n * (seq_id x).n by SEQ_1:8;
    hence thesis by XREAL_1:63;
  end;
A2: seq_id(x)(#)seq_id(x) is summable & x .|. x = Sum(seq_id(x)(#)seq_id(x)
  ) by Th1;
A3: now
A4: for n be Nat holds 0 <= (seq_id(x)(#)seq_id(x)).n
    proof
      let n be Nat;
      (seq_id(x)(#)seq_id(x)).n =(seq_id(x)).n * (seq_id(x)).n by SEQ_1:8;
      hence thesis by XREAL_1:63;
    end;
    assume
A5: x .|. x = 0;
A6: x .|. x = Sum(seq_id(x)(#)seq_id(x)) & seq_id(x)(#)seq_id(x) is
    summable by Th1;
A7: for n be Nat holds 0 = (seq_id x).n
    proof
      let n be Nat;
      0 = (seq_id(x)(#)seq_id(x)).n by A5,A4,A6,RSSPACE:17
        .= ((seq_id(x)).n)*((seq_id(x)).n) by SEQ_1:8;
      hence thesis by XCMPLX_1:6;
    end;
    x is Element of the_set_of_RealSequences by RSSPACE:def 11;
    hence x=0.l2_Space by A7,Th1,RSSPACE:5;
  end;
A8: x .|. y = Sum(seq_id(x)(#)seq_id(y)) by Th1
    .=y .|. x by Th1;
A9: now
    assume
A10: x=0.l2_Space;
A11: for n be Nat holds (seq_id(x)(#)seq_id(x)).n=0
    proof
      let n be Nat;
      thus (seq_id(x)(#)seq_id(x)).n = (seq_id(x)).n * (seq_id(x)).n by SEQ_1:8
        .=(seq_id(x)).n * 0 by A10,Th1
        .= 0;
    end;
    thus x .|. x = Sum(seq_id(x)(#)seq_id(x)) by Th1
      .=0 by A11,RSSPACE:16;
  end;
A12: seq_id(x)(#)seq_id(y) is summable by Th1;
A13: (a*x) .|. y =Sum(seq_id(a*x)(#)seq_id(y)) by Th1
    .=Sum(a(#)seq_id(x)(#)seq_id(y)) by Th1
    .=Sum(a(#)(seq_id(x)(#)seq_id(y))) by SEQ_1:18
    .=a*Sum(seq_id(x)(#)seq_id(y)) by A12,SERIES_1:10
    .=a * ( x .|. y ) by Th1;
A14: seq_id(x)(#)seq_id(z) is summable & seq_id(y)(#)seq_id(z) is summable
  by Th1;
  (x+y) .|. z = Sum(seq_id(x+y)(#)seq_id(z)) by Th1
    .= Sum((seq_id(x)+seq_id(y))(#)seq_id(z)) by Th1
    .= Sum(seq_id(x)(#)seq_id(z)+seq_id(y)(#)seq_id(z)) by SEQ_1:16
    .= Sum(seq_id(x)(#)seq_id(z))+Sum(seq_id(y)(#)seq_id(z)) by A14,SERIES_1:7
    .= x .|. z + Sum(seq_id(y)(#)seq_id(z)) by Th1
    .= x .|. z + y .|. z by Th1;
  hence thesis by A3,A9,A1,A2,A8,A13,SERIES_1:18;
end;
