reserve u,v,x,y,z,X,Y for set;
reserve r,s for Real;
reserve N for non empty ConjNormAlgStr;
reserve a,a1,a2,b,b1,b2 for Element of N;

theorem Th10:
  N is well-conjugated reflexive discerning RealNormSpace-like
  vector-distributive scalar-distributive scalar-associative scalar-unital
  Abelian add-associative right_zeroed right_complementable associative
  distributive well-unital non degenerated almost_left_invertible
  implies (-a)*' = -(a*')
  proof
    assume that
A1: N is well-conjugated and
A2: N is reflexive discerning RealNormSpace-like
    vector-distributive scalar-distributive scalar-associative scalar-unital
    Abelian add-associative right_zeroed right_complementable and
A3: N is associative and
A4: N is distributive and
A5: N is well-unital and
A6: N is non degenerated almost_left_invertible;
    per cases;
    suppose
A7:   a is non zero;
      then
A8:   a*' * a = ||.a.||^2 * 1.N by A1,Def3;
A9:   (-a)*' * -a = ||.-a.||^2 * 1.N by A1,A2,A6,A7,Def3;
A10:   ||.-a.|| = ||.a.|| by A2,NORMSP_1:2;
A11:   a <> 0.N by A7;
A12:   a*/a = 1.N by A2,A3,A4,A5,A6,A11,VECTSP_2:9;
      then
A13:   (-a) * -/a = 1.N by A2,A4,VECTSP_1:10;
A14:   a * -/a = -1.N by A2,A4,A12,VECTSP_1:8;
      thus (-a)*' = (-a)*' * 1.N by A5
      .= (-a)*' * (-a) * -/a by A13,A3
      .= a*' * (a * -/a) by A8,A9,A10,A3
      .= -(a*') by A2,A14,A4,A5,VECTSP_2:28;
    end;
    suppose a is zero;
      then
A15:   a = 0.N & a*' = 0.N by A1;
      -0.N = 0.N by A2;
      hence thesis by A15;
    end;
  end;
