
theorem rc:
for X being non empty set
for L being non empty
            add-associative right_zeroed right_complementable addLoopStr
for f being Function of X,L holds f '+' ('-' f) = X --> 0.L
proof
let X be non empty set,
L be non empty add-associative right_zeroed right_complementable addLoopStr,
f be Function of X,L;
now let o be object;
  assume o in X;
  then reconsider x = o as Element of X;
  thus (f '+' ('-' f)).o
     = f.x + ('-'f).x by defp
    .= f.x + -(f.x) by defm
    .= (X --> 0.L).o by RLVECT_1:5;
  end;
hence thesis by FUNCT_2:12;
end;
