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
  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
  add-conjugative
  implies (a-b)*' = a*' - b*'
  proof
    assume that
A1: 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;
    assume N is add-conjugative;
    hence (a-b)*' = a*'+(-b)*'
    .= a*'-b*' by A1,Th10;
  end;
