reserve X for non empty CUNITSTR;
reserve a, b for Complex;
reserve x, y for Point of X;
reserve X for ComplexUnitarySpace;
reserve x, y, z, u, v for Point of X;
reserve seq, seq1, seq2, seq3 for sequence of X;
reserve  n for Nat;

theorem Th74:
  for l be CLSStruct st the CLSStruct of l is ComplexLinearSpace
  holds l is ComplexLinearSpace
proof
  let l be CLSStruct such that
A1: the CLSStruct of l is ComplexLinearSpace;
  reconsider l as non empty CLSStruct by A1;
  reconsider l0=CLSStruct (# the carrier of l, 0.l, the addF of l, the Mult of
    l #) as ComplexLinearSpace by A1;
A2: l is Abelian
  proof
    let v,w be VECTOR of l;
    reconsider v1=v as VECTOR of l0;
    reconsider w1=w as VECTOR of l0;
    thus v+w =v1+w1 .= w1+v1
      .= w +v;
  end;
A3: l is add-associative
  proof
    let u,v,w be VECTOR of l;
    reconsider u1=u as VECTOR of l0;
    reconsider v1=v as VECTOR of l0;
    reconsider w1=w as VECTOR of l0;
    thus (u + v) + w = (u1+v1)+w1 .= u1+(v1+w1) by RLVECT_1:def 3
      .= u+(v+w);
  end;
A4: l is vector-distributive scalar-distributive scalar-associative
scalar-unital
  proof
    thus for z being Complex, v,w being VECTOR of l holds z * (v + w) = z * v
    + z * w
    proof
      let z be Complex;
      let v,w be VECTOR of l;
      reconsider v1=v, w1=w as VECTOR of l0;
      thus z*(v+w) =z*(v1+w1) .=z*v1+z*w1 by CLVECT_1:def 2
        .= z*v +z*w;
    end;
    thus for z1,z2 being Complex, v being VECTOR of l holds (z1 + z2) * v = z1
    * v + z2 * v
    proof
      let z1,z2 be Complex;
      let v be VECTOR of l;
      reconsider v1=v as VECTOR of l0;
      thus (z1+z2)*v =(z1+z2)*v1 .=z1*v1+z2*v1 by CLVECT_1:def 3
        .= z1*v +z2*v;
    end;
    thus for z1,z2 being Complex, v being VECTOR of l holds (z1 * z2) * v = z1
    * (z2 * v)
    proof
      let z1,z2 be Complex;
      let v be VECTOR of l;
      reconsider v1=v as VECTOR of l0;
      thus (z1*z2)*v =(z1*z2)*v1 .=z1*(z2*v1) by CLVECT_1:def 4
        .= z1*(z2*v);
    end;
    let v be VECTOR of l;
    reconsider v1=v as VECTOR of l0;
    thus 1r*v= 1r*v1 .= v by CLVECT_1:def 5;
  end;
A5: l is right_complementable
  proof
    let v be VECTOR of l;
    reconsider v1=v as VECTOR of l0;
    consider w1 being VECTOR of l0 such that
A6: v1 + w1 = 0.l0 by ALGSTR_0:def 11;
    reconsider w = w1 as VECTOR of l;
    take w;
    thus thesis by A6;
  end;
  l is right_zeroed
  proof
    let v be VECTOR of l;
    reconsider v1=v as VECTOR of l0;
    thus v+0.l= v1 + 0.l0 .= v by RLVECT_1:def 4;
  end;
  hence thesis by A2,A3,A5,A4;
end;
