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 Th18:
  y<>0.SF & z<>0.SF implies x/y=(x*z)/(y*z)
proof
  assume
A1: y<>0.SF;
  assume
A2: z<>0.SF;
  thus x/y=x*1_SF*y"
    .=x*(z*z")*y" by A2,Th9
    .=(x*z)*z"*y" by GROUP_1:def 3
    .=(x*z)*(z"*y") by GROUP_1:def 3
    .=(x*z)/(y*z) by A1,A2,Th11;
end;
