reserve FS for non empty doubleLoopStr;
reserve F for Field;
reserve R for Abelian add-associative right_zeroed right_complementable non
  empty addLoopStr,
  x, y, z for Scalar of R;
reserve SF for Skew-Field,
  x, y, z for Scalar of SF;

theorem Th19:
  y<>0.SF implies -x/y=(-x)/y & x/(-y)=-x/y
proof
  assume y<>0.SF;
  then
A1: -y<>0.SF by Th3;
  thus
A2: -x/y=(-x)/y by VECTSP_1:9;
  -1.SF<>0.SF by Th3;
  then x/(-y)=(x*(-1_SF))/((-y)*(-1_SF)) by A1,Th18;
  then x/(-y)=(-x*1_SF)/((-y)*(-1_SF)) by VECTSP_1:8
    .= (-x)/((-y)*(-1_SF))
    .= (-x)/(-(-y)*1_SF) by VECTSP_1:8
    .= (-x)/((-(-y))*1_SF)
    .= (-x)/(y*1_SF) by RLVECT_1:17
    .= -x/y by A2;
  hence thesis;
end;
