reserve x,X,Y for set;
reserve C for non empty set;
reserve c for Element of C;
reserve V for RealNormSpace;
reserve f,f1,f2,f3 for PartFunc of C,V;
reserve r,r1,r2,p for Real;

theorem Th23:
  for V being add-associative right_zeroed right_complementable Abelian
    scalar-distributive scalar-unital vector-distributive
    non empty RLSStruct
  for f being PartFunc of C,V holds
  -f = (-1)(#)f
proof
  let V be add-associative right_zeroed right_complementable Abelian
  scalar-distributive scalar-unital vector-distributive
  non empty RLSStruct;
  let f be PartFunc of C,V;
A1: dom (-f) = dom f by Def5
    .= dom ((-1)(#)f) by Def4;
  now
    let c;
    assume
A2: c in dom ((-1)(#)f);
    hence (-f)/.c = -(f/.c) by A1,Def5
      .= (-1) * f/.c by RLVECT_1:16
      .= ((-1)(#)f)/.c by A2,Def4;
  end;
  hence thesis by A1,PARTFUN2:1;
end;
